arXiv:2108.05912v2 [math.AG] 14 Mar 2022

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LOCAL TROPICALIZATIONS OF SPLICE TYPE SURFACE SINGULARITIES

MARIA ANGELICA CUETO, PATRICK POPESCU-PAMPU§ AND DMITRY STEPANOV(WITH AN APPENDIX BY JONATHAN WAHL)

To Walter Neumann, on the occasion of his 75th birthday.

Abstract. Splice type surface singularities were introduced by Neumann and Wahl as a generalization of theclass of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on aweighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint.We characterize their local tropicalizations as the cones over the appropriately embedded associated splicediagram. Furthermore, under suitable coprimality conditions on the weights of the splice diagram, we showthat the diagram can be uniquely recovered from the local tropicalization. As a corollary, we reprove someof Neumann and Wahl’s earlier results on these singularities by purely tropical methods, and show thatsplice type surface singularities are Newton non-degenerate in the sense of Khovanskii. Finally, we use thesetropical fans to exhibit explicit toric morphisms resolving these singularities.

1. Introduction

Splice diagrams are finite trees with half-edges weighted by integers and with nodes (internal vertices)decorated by ± signs. If the half-edge weights around each node are pairwise coprime, we say the splicediagram is coprime. This class of weighted trees was first introduced by Siebenmann [46] in 1980 to encodegraph manifolds which are integral homology spheres. Coprime splice diagrams with only + node decora-tions and positive half-edge weights were used by Eisenbud and Neumann in [6] to study normal surfacesingularities. One of the main theorems of [6] states that integral homology sphere links of normal surfacesingularities are described by positively-weighted coprime splice diagrams satisfying the edge determinantcondition, namely, that the product of the two weights associated to any fixed internal edge must be greaterthan the product of the weights of the neighboring half-edges.

Interesting isolated surface singularities arise from splice diagrams. For example, complete intersectionsof Pham-Brieskorn-Hamm hypersurface singularities are associated to star splice diagrams (i.e., those with asingle node). As recognized by Hamm in [15, §5] and [16], in order to determine an isolated singularity in Cn,all maximal minors of the coefficient matrix (cij)i,j of each polynomial fi :=

∑

j cijzaj

j in the Brieskorn system

must be non-zero. In turn, work of Neumann [29] shows that universal abelian covers of quasi-homogeneouscomplex normal surface singularities with rational homology sphere links are complete intersections of Pham-Brieskorn-Hamm hypersurface singularities.

In 2002, Neumann and Wahl [31] extended this family of complete intersections by defining splice typesurface singularities associated to splice diagrams whose weights satisfy a special arithmetic property calledthe semigroup condition. These singularities are defined by explicit splice type systems of convergent powerseries near the origin, whose coefficients satisfy generalizations of Hamm’s maximal minors conditions. Splicetype surface singularities and the related class of splice quotients have been further studied by both authorsin [32, 33, 34], and by Lamberson, Nemethi, Okuma and Pedersen in [22, 25, 26, 27, 36, 37, 38, 40, 41, 42].For more details, we refer the reader to the surveys [30, 39, 52, 53].

The present paper uses tropical geometry techniques to study splice type systems with n leaves associatedto splice diagrams satisfying the edge determinant and semigroup conditions. Our first main result recoversand strengthens a central theorem from [32] (see Theorem 2.15). More precisely:

Theorem 1.1. Splice type systems are Newton non-degenerate complete intersection systems of equations.The associated splice type singularities are isolated, irreducible and not contained in any coordinate subspaceof the corresponding ambient space Cn.

2020 Mathematics Subject Classification. Primary : 14B05, 14T90, 32S05; Secondary : 14M25, 57M15.Key words and phrases. Surface singularities, complete intersection singularities, tropical geometry, Newton non-degeneracy.§ Corresponding author .

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http://arxiv.org/abs/2108.05912v2

2 M.A. CUETO, P. POPESCU-PAMPU, D. STEPANOV

This statement plays a mayor role in the proof of the Neumann-Wahl Milnor fiber conjecture for splice typesingularities with integral homology sphere links [33] obtained by the three authors. For an overview of thisproof, we refer the reader to [5].

The notion of Newton non-degeneracy (in the sense of Kouchnirenko [20] and Khovanskii [17]) is closelyrelated to the notion of initial form of a series relative to a weight vector, which lies at the core of tropicalgeometry. A regular sequence of convergent power series in C{z1, . . . , zn} defining a germ (X, 0) → Cn isa Newton non-degenerate complete intersection system if for each weight vector w with positive entries, theassociated initial forms (inw(fi))i determine hypersurfaces of the algebraic torus (C∗)n whose sum is a normalcrossings divisor in the neighborhood of their intersection. This condition is automatically satisfied wheneverthe intersection is empty. Surprisingly, not many examples of Newton non-degenerate complete intersectionsystems are known in codimension two or higher. Theorem 1.1 contributes a large class of examples of suchsystems.

Newton non-degeneracy enables the resolution of the germ (X, 0) → Cn by a single toric morphism.Indeed, works of Varchenko [51] (for hypersurfaces) and Oka [35, Chapter III, Theorem (3.4)] (for completeintersections) show that such a morphism may be defined by a regular subdivision of the positive orthantrefining the dual fan of the Newton polyhedron of each function fi. Furthermore, the complete dual fan is notneeded to achieve a resolution. Indeed, [35, Theorem III.3.4] allows us to restrict to the subfan correspondingto the orbits intersecting the strict transform of the given germ.

The support of this subfan depends only on the ideal defining the germ (X, 0) but not on the particulargenerators fi. It is the finite part of the so-called local tropicalization of the embedding (X, 0) → Cn. Thislocal version of the standard notion of tropicalization of a subvariety of an algebraic torus was first introducedby the last two authors in [43] as a tool to study arbitrary subgerms of Cn or, more generally, arbitrarymorphisms from analytic germs to germs of toric varieties.

As was mentioned earlier, splice diagrams record topological information about the link of splice typesingularities. Indeed, starting from a normal surface singularity with a rational homology sphere link,Neumann and Wahl [31] build a splice diagram which determines the dual tree of the minimal normalcrossings resolution (up to a finite ambiguity) whenever this graph is not a star tree. This diagram ishomeomorphic to the dual tree: it is obtained by disregarding all bivalent vertices of the dual tree. Itsatisfies the edge determinant condition, but not necessarily the semigroup or the congruence conditions.When (X, 0) → Cn has an integral homology sphere link, the ambiguity disappears and the dual tree iscompletely determined by the splice diagram.

Given a splice diagram Γ with n leaves, the construction of Neumann and Wahl associates a weight vectorwu ∈ (Z>0)

n to each vertex u of Γ. These vectors induce a piecewise linear embedding of Γ into the standardsimplex in Rn after appropriate normalization. Our second main result shows the close connection betweenΓ, the local tropicalization of the associated splice type system in Cn and its resolution diagrams:

Theorem 1.2. Let Γ be a splice diagram satisfying the semigroup condition and let (X, 0) → Cn be thegerm defined by an associated splice type system. Then, the finite local tropicalization of X is the cone overan embedding of Γ in Rn. Furthermore, in the coprime case, Γ can be uniquely recovered from this fan.

Theorem 1.2 shows that the link at the origin of the local tropicalization of (X, 0) → Cn (obtained byintersecting the fan with the (n− 1)-dimensional sphere) is homeomorphic to the splice diagram Γ. To thebest of our knowledge, this is the first tropical interpretation of Siebenmann’s splice diagrams. In this spirit,we view Siebenmann’s paper [46] as a precursor to tropical geometry (for others, see [23, Chapter 1]).

Our method to characterize the local tropicalization is different from the general one discussed in Oka’sbook [35] and described briefly above. Namely, we do not use the Newton polyhedra of the collection of seriesdefining the germ. Instead, we use a “mine-sweeping” approach, using successive stellar subdivisions of thestandard simplex in Rn dictated by the splice diagram, in order to remove relatively open cones in the positiveorthant avoiding the local tropicalization. Once the local tropicalization is determined (via Theorem 1.2),a simple computation confirms the Newton non-degeneracy of the system. In turn, by analyzing the localtropicalizations of the intersections of the germ with the coordinate subspaces of Cn, we conclude that (X, 0)is an isolated complete intersection surface singularity, thus completing the proof of Theorem 1.1.

As a consequence of Theorem 1.1, we provide an alternative proof of the main theorem of de Felipe,Gonzalez Perez and Mourtada [8], stating that any germ of a reduced plane curve may be resolved by onetoric modification after re-embedding its ambient smooth germ of surface into a higher dimensional germ

LOCAL TROPICALIZATIONS OF SPLICE TYPE SURFACE SINGULARITIES 3

(Cn, 0) (see Corollary 7.11). The first theorem of this kind was proved by Goldin and Teissier [14] forirreducible germs of plane curves.

Our paper is organized as follows. In Section 2, we review the definitions and main properties of splicediagrams, splice type systems and end-curves associated to rooted splice diagrams, following [32, 33]. Sec-tions 3 and 4 include background material of local tropicalizations and Newton non-degeneracy. In Section 5we show how to embed a given splice diagram Γ with n leaves into the standard (n− 1)-simplex in Rn, andwe highlight various convexity properties of this embedding. The proof of the first part of Theorem 1.2 isdiscussed in Section 6, while Theorem 1.1 is proven in Section 7. Section 8 characterizes local tropicaliza-tions of splice type systems defined by a coprime splice diagram and shows how to recover the diagram fromthe tropical fan, thus yielding the second part of Theorem 1.2. Finally, Section 9 discusses the dependency ofthe construction of splice type systems on the choice of admissible monomials for arbitrary splice diagrams,in the spirit of [32, Section 10].

Appendix A, written by Jonathan Wahl, includes a proof of [32, Lemma 3.3] that was absent from theliterature. This result confirms that given a finite sequence (f1, . . . , fs) in C{z1, . . . , zn} and a fixed positiveinteger vector w ∈ Zn, the regularity of the sequence (inw(f1), . . . , inw(fs)) of initial forms ensures that theoriginal sequence is regular, and furthermore, the w-initial ideal must be generated by the sequence of initialforms. This statement is essential to determine if a given w lies in the local tropicalization of the germdefined by the vanishing of the input sequence (see Corollary 7.5), providing an alternative proof to partof Theorem 1.2.

2. Splice diagrams, splice type systems and end-curves from rooted splice diagrams

In this section, we recall the notions of splice diagram and splice type systems associated to them. Thedefinitions follow closely the work of Neumann and Wahl in [32, 33].

We start with some basic terminology and notations about trees:

Definition 2.1. A tree is a connected graph with no cycles and at least one vertex. The star of a vertex v

of the tree Γ is the collection StarΓ(v) of edges adjacent to v. The valency of v is the cardinal of StarΓ(v),

which we denote by δv . A node of a tree is a vertex v whose valency is greater than one, whereas a leaf is

a one-valent vertex. We denote the set of nodes of the tree Γ by V ◦(Γ) and its set of leaves by ∂ Γ .

When the ambient tree is understood from the context, we remove it from the notation and simply writeStar(v).

Definition 2.2. Given a subsetW = {p1, . . . , pk} of vertices of the tree Γ, we denote by [W ] or [p1, . . . , pk]

the subtree of Γ spanned by these points. We call it the convex hull of the set {p1, . . . , pk} inside Γ. Forexample, Γ = [∂ Γ].

Splice diagrams are special kinds of trees enriched with weights around all nodes, as we now describe:

Definition 2.3. A splice diagram is a pair (Γ, {dv,e}v,e), where Γ is a tree without valency-two vertices,with at least one node and which is decorated with a weight function on the star of each node v of Γ, denotedby

StarΓ(v)→ Z>0 : e 7→ dv,e.

We call dv,e the weight of e at v. If u is any other vertex of Γ such that e lies in the unique geodesic of

Γ joining u and v, we write dv,u := dv,e. We view this as the weight in the neighborhood of v pointing

towards u. The total weight of a node v of Γ is the product dv :=∏

e∈StarΓ(v)

dv,e.

Remark 2.4. Let (Γ, {dv,e}v,e) be a splice diagram. For simplicity, we remove the collection of weights fromthe notation and simply use Γ to refer to the splice diagram. By a similar abuse of notations, we may viewStarΓ(v) also as a splice diagram, whose weights around its unique node v are inherited from Γ. Splicediagrams with one node will be referred to as star splice diagrams, and the underlying graphs as star trees.


Definition 2.5. Let u and v be two distinct vertices of the splice diagram Γ. The linking number ℓu,v

between u and v is the product of all the weights adjacent to, but not on, the geodesic [u, v] joining u and v.

Thus, ℓv,u = ℓu,v. We set ℓv,v := dv for each node v of Γ. The reduced linking number ℓ′u,v is defined via

a similar product where we exclude the weights around u and v. In particular, ℓ′v,v = 1 for each node v of Γ.

Remark 2.6. Given a node v and a leaf λ of Γ, it is immediate to check that ℓv,λ dv,λ = ℓ′v,λ dv.

Linking numbers satisfy the following useful identity, whose proof is immediate from Definition 2.5 (see[10, Proposition 69]):

Lemma 2.7. If u, v, w are vertices of Γ with u ∈ [v, w], then ℓu,v ℓu,w = du ℓv,w.

In [28, Theorem 1], Neumann gave explicit descriptions of integral homology spheres associated to starsplice diagrams as links of Pham-Brieskorn-Hamm surface singularities. The following definition was in-troduced by Neumann and Wahl in [33, Sect. 1] to characterize which integral homology spheres may berealized as links of normal surface singularities. Its origins can be traced back to [6, page 82].

Definition 2.8. Let Γ be a splice diagram. Given two adjacent nodes u and v of Γ, the determinant ofthe edge [u, v] is the difference between the product of the two decorations on [u, v] and the product of theremaining decorations in the neighborhoods of u and v, that is,

(2.1) det([u, v]) := du,vdv,u − ℓu,v.

The splice diagram Γ satisfies the edge determinant condition if all edges have positive determinants.

As was shown by Eisenbud and Neumann in [6, Thm. 9.4], the edge determinant condition characterizeswhich integral homology spheres can be realized as surface singularity links. More precisely,

Theorem 2.9. The integral homology sphere links of normal surface singularities are precisely the oriented3-manifolds Σ(Γ) associated to splice diagrams Γ which satisfy the edge determinant condition.

The construction of oriented 3-manifolds from splice diagrams is due to Siebenmann [46]. They are obtainedfrom splicing Seifert-fibered oriented 3-manifolds Σ(StarΓ(v)) associated to each node v of Γ along specialfibers of their respective Seifert fibration corresponding to the edges of Γ. For each Σ(StarΓ(v)), thesespecial fibers are in bijection with the δv-many edges adjacent to v. Each edge [u, v] induces a splicing ofboth Σ(StarΓ(u)) and Σ(StarΓ(v)) along the oriented fibers corresponding to the edge. These fibers areknots in both Seifert-fibered manifolds and their linking number is precisely ℓu,v (see [6, Thm. 10.1]).

The following result shows that the edge determinant condition yield Cauchy-Schwarz’ type inequalities:

Lemma 2.10. Assume that the splice diagram Γ satisfies the edge determinant condition. Then,

(2.2) du dv ≥ ℓ2u,v, for all nodes u, v ∈ Γ.

Furthermore, equality holds if and only if u = v.

Proof. The result follows by induction on the distance distΓ(u, v) between u and v. The base case correspondsto adjacent nodes. Lemma 2.7 is used for the inductive step. �

If Γ satisfies the edge determinant condition, then the linking numbers verify the following inequality,which generalizes Lemma 2.7 and is reminiscent of the ultrametric condition for dual graphs of arborescentsingularities. For a precise statement, we refer the reader to [13, Proposition 1.18]:

Proposition 2.11. Assume that the splice diagram Γ satisfies the edge determinant condition. Then, forall nodes u, v and w of Γ, we have ℓu,v ℓu,w ≤ du ℓv,w. Furthermore, equality holds if and only if u ∈ [v, w].

Proof. Consider the tree T spanned by u, v and w and let a be the unique node in the intersections of thethree geodesics [u, v], [u,w] and [v, w]. We prove the inequality by a direct calculation. By Lemma 2.7applied to the triples {u, v, a}, {u,w, a} and {v, w, a}, we have:

ℓu,a ℓa,v = ℓu,v da, ℓu,a ℓa,w = ℓu,w da and ℓw,a ℓa,v = ℓw,v da.

These expressions combined with the inequality (2.2) applied to the pair {a, u} yield:

(2.3) ℓu,v ℓu,w =ℓv,a ℓa,u

da

ℓw,a ℓa,uda

=ℓv,a ℓw,a

da

(ℓa,u)2

da= ℓw,v

(ℓa,u)2

da≤ ℓw,v du.


Furthermore, Lemma 2.10 confirms that equality is attained if and only if a = u, that is, if and only if u liesin the geodesic [v, w]. This concludes our proof. �

Before introducing splice type systems as defined by Neumann and Wahl in [32, 33], we set up notationarising from toric geometry. We write n := |∂ Γ| for the number of leaves of the splice diagram Γ (where

| | denotes the cardinality of a finite set) and let M(∂ Γ) be the free abelian group generated by all leaves

of Γ. We denote by N(∂ Γ) its dual lattice and write the associated pairing using dot product notation,

i.e. w ·m whenever w ∈ N(∂ Γ) and m ∈ M(∂ Γ). Fixing a basis { wλ : λ ∈ ∂ Γ} for N(∂ Γ) and its dual

basis { mλ : λ ∈ ∂ Γ} for M(∂ Γ) identifies both lattices with Zn. To each mλ, we associated a variable

zλ := χmλ . We view M(∂ Γ) as the lattice of exponents of monomials in those variables and N(∂ Γ) as the

associated lattice of weight vectors.In addition to defining weights for all leaves of Γ, each node v in Γ has an associated weight vector:

(2.4) wv :=∑

λ∈∂ Γ

ℓv,λ wλ ∈ N(∂ Γ).

As was mentioned in Section 1, star splice diagrams Γ with a unique node v produce Pham-Brieskorn-

Hamm singularities using the monomials {zdv,λ

λ : λ ∈ ∂ Γ}. Neumann and Wahl’s splice type systems [32, 33]generalize this construct to diagrams with more than one node. In addition to satisfying the edge determinant

condition, Γ must have an extra arithmetic property that allows to replace each monomial zdv,λv by a suitable

monomial associated to the pair (v, e) where v is any node and e is an edge adjacent to it (see (2.8)).This property, which ensures that all monomials associated to a vertex v have the same wv-degree, willautomatically hold for star splice diagrams.

Definition 2.12. A splice diagram Γ satisfies the semigroup condition if for each node v and each edgee ∈ Star(v), the total weight dv of v belongs to the subsemigroup of (N,+) generated by the set of linkingnumbers between v and the leaves λ seen from v in the direction of e, that is, such that e ⊆ [v, λ]. We write

(2.5) dv =∑

λ∈∂v,eΓ

mv,e,λ ℓv,λ , or equivalently dv,e =∑

λ∈∂v,eΓ

mv,e,λ ℓ′v,λ,

where mv,e,λ ∈ N for all λ and ∂v,eΓ is the set of leaves λ of Γ with e ⊆ [v, λ].

We use the coefficients from (2.5) to define an exponent vector (i.e., an element of M(∂ Γ)) for each pair(v, e) as above:

(2.6) mv,e :=∑

λ∈∂v,eΓ

mv,e,λ mλ ∈M(∂v,eΓ) ⊂M(∂ Γ).

Following [33], we refer to it as an admissible exponent for (v, e). By (2.5) it satisfies

(2.7) wv ·mv,e = dv for each edge e ∈ Star(v).

Each admissible exponent mv,e defines an admissible monomial, which was denoted by Mv,e in [33]:

(2.8) zmv,e :=∏

λ∈∂v,eΓ

zmv,e,λ

λ .

Definition 2.13. Let Γ be a splice diagram which satisfies both the edge determinant and the semigroupconditions of Definitions 2.8 and 2.12. We fix an order for its set of n leaves ∂ Γ.

• A strict splice type system associated to Γ is a finite family of (n− 2) polynomials of the form:

(2.9) fv,i(z) :=∑

e∈Star(v)

cv,e,i zmv,e for all i ∈ {1, . . . , δv − 2} and v a node of Γ,

where mv,e ∈ M(∂Γ) are the admissible exponent vectors defined in (2.6). We also require thecoefficients cv,e,i to satisfy the Hamm determinant conditions. Namely, for any node v ∈ Γ, if wefix an ordering of the edges in Star(v), then all the maximal minors of the matrix of coefficients(cv,e,i)e,i ∈ Cδv×(δv−2) must be non-zero.


Figure 1. From left to right: a splice diagram and its associated rooted diagram obtainedby fixing one of the leaves as its root r, and removing one weight from the star of each node.

• A splice type system S(Γ) associated to Γ is a finite family of power series of the form

(2.10) Fv,i(z) := fv,i(z) + gv,i(z) for all i ∈ {1, . . . , δv − 2} and v a node of Γ,

where the collection (fv,i)v,i is a strict splice type system associated to Γ and each gv,i is a convergentpower series satisfying the following condition for each exponent m in the support of gv,i (comparewith the equality imposed in (2.7)):

(2.11) wv ·m > dv and wu ·m > ℓu,v for each node u of Γ with u 6= v.

• A splice type singularity associated to Γ is the germ at Cn defined by S(Γ).Remark 2.14. As was shown by Neumann and Wahl in [32, Lemma 3.2], the right-most inequality in (2.11)follows from the left-most one and the edge determinant condition. We choose to include both inequalitiesin (2.11) for mere convenience since we will need both of them for several arguments in Section 6.

The issue of dependency of the set of germs defined by splice type systems on the choice of admissiblemonomials is a subtle one. We postpone this discussion to Section 9.

As was mentioned in Section 1, splice type singularities satisfy the following crucial property, proved byNeumann and Wahl in [32, Thm. 2.6]. An alternative proof of this statement, using local tropicalization,will be provided at the end of Section 7.

Theorem 2.15. Splice type singularities are isolated complete intersection surface singularities.

Example 2.16. We let Γ be the splice diagram to the left of Figure 1. Then, du = 294, dv = 770 andℓu,v = 420 and so the edge determinant condition du dv > ℓ2u,v holds for [u, v]. Furthermore, since

49 = 0 · (2 · 5) + 1 · (2 · 7) + 1 · (5 · 7) and 11 = 1 · (3) + 4 · (2) = 3 · (3) + 1 · (2),the semigroup condition is also satisfied and one may take as exponents mu,[u,v] = (0, 0, 0, 1, 1) and mv,[u,v] =

(1, 4, 0, 0, 0) or (3, 1, 0, 0, 0) in Z5. A possible strict splice type system for Γ is:

(2.12)

fu,1 := z21 − 2 z32 + z4 z5,

fv,1 := z1z42 + z73 + z54 − 2155 z25 ,

fv,2 := 33 z1z42 + z73 + 2 z54 − 2123 z25 .

An alternative system is obtained by replacing the admissible monomial z1z42 with z31z2. The coefficients

of the system were chosen to simplify the parameterization of the end-curve of the associated splice typesurface singularity, associated to the leaf λ1 (see Example 2.23 for details). ⋄

A central role in this paper will be played by tropicalizations and weighted initial forms of series andideals of C[zλ : λ ∈ ∂ Γ], which we discuss in Section 3. In Proposition 2.19, we determine the initial formsof the series Fv,i of a splice type system S(Γ) with respect to each weight vector wu from (2.4). Its proof isa consequence of the next two lemmas:

Lemma 2.17. Assume that Γ is a splice diagram satisfying the edge determinant condition. Then, for anypair of adjacent nodes u, v of Γ we have:

wu ∈ℓu,vdv

wv + R≥0〈wλ : λ ∈ ∂v,[u,v]Γ〉.


Proof. We write e = [u, v]. The definition of linking numbers gives the following expressions for each λ ∈ ∂ Γ:

ℓu,λ =

{

(ℓv,λ du)/(dv,u du,v) if λ ∈ ∂u,eΓ,

(ℓv,λ dv,u du,v)/dv if λ ∈ ∂v,eΓ.

The statement follows by substituting these expressions in the definition of wu from (2.4) and by using theedge determinant condition, i.e.

�(2.13) wu =du

dv,u du,v

∑

λ∈∂u,eΓ

ℓv,λ wλ +dv,u du,v

dv

∑

λ∈∂v,eΓ

ℓv,λ wλ =ℓu,vdv

wv +∑

λ∈∂v,eΓ

det(e) ℓv,λdv

︸︷︷︸

>0

wλ.

Lemma 2.18. Assume that Γ satisfies the edge determinant and semigroup conditions. Then, the exponentvector mv,e from (2.6) satisfies wu ·mv,e ≥ ℓu,v for all nodes u of Γ and each edge e ∈ StarΓ(v). Furthermore,equality holds if and only if e 6⊆ [u, v].

Proof. If u = v, then wv · mv,e = dv = ℓv,v. If u 6= v, we argue by induction on distΓ(u, v) > 0. IfdistΓ(u, v) = 1, we let e′ = [u, v]. Expression (2.13) yields

wu ·mv,e =ℓu,vdv

wv ·mv,e +det(e′)

dv

( ∑

λ∈∂v,e′Γ

mv,e,λ ℓv,λ)= ℓu,v +

det(e′)

dv

( ∑

λ∈∂v,e′Γ

mv,e,λ ℓv,λ).

The second summand is always non-negative and it equals zero if and only if e 6= e′.If distΓ(u, v) > 1, we let u′ be the unique node adjacent to u in [u, v] and set e′ := [u′, u]. Note that

e ∈ [v, u] if and only if e ∈ [v, u′]. Expression (2.13) applied to {u, u′}, the non-negativity of each mv,e,λ, theinductive hypotheses on {u′, v} and Proposition 2.11 yield

wu ·mv,e =ℓu,u′

du′

wu′ ·mv,e︸︷︷︸

≥ℓu′,v

+det(e′)

du′

∑

λ∈∂u′,e′Γ

mv,e,λℓu′,λ︸︷︷︸

≥0

≥ ℓu,u′ ℓu′,v

du′

= ℓu,v.

By construction, equality is achieved if and only if e ⊆ [u′, v], which is equivalent to e ⊆ [u, v]. �

As an immediate consequence of the previous lemma, we can determine the wu-initial form of each seriesin S(Γ) (see Definition 3.1):

Proposition 2.19. For each pair of nodes u, v in Γ, we have:

inwu(Fv,i) =

{

fu,i if u = v,

fv,i − cv,[u,v] zmv,e otherwise,

where e is the unique edge adjacent to v with e ⊆ [v, u].

The second formula, for the case where u 6= v, means that we remove from fv,i the term associated to theedge starting from v in the direction of u.

Example 2.20. Given the minimal splice type system from Example 2.16, we havewu = (147, 98, 60, 84, 210)and wv = (210, 140, 110, 154, 385). The polynomial fu,1 is wu-homogeneous, whereas fv,1 and fv,2 are wv-homogeneous. Their weighted initial parts relative to wu and wv are:

inwv(fu,1) = fu,1 − z4z5 = z21 − 2 z32 ,

inwu(fv,1) = fv,1 − z1z

42 = z73 + z54 − 2155 z25 ,

inwu(fv,2) = fv,2 − 33 z1z

42 = z73 + 2 z54 − 2123 z25 .

⋄We will be interested in curves obtained from a given splice type system when we choose a leaf r of the

corresponding splice diagram Γ to be its root. We orient the resulting rooted tree Γr towards the root and

remove one weight in the neighborhood of each node, namely the one pointing towards the root, as seen on

the right of Figure 1. We write ∂rΓ for the set of (n − 1) non-root leaves of Γr and assume it is ordered.

The following definition was introduced in [32, Section 3] by the name of splice diagram curves. In viewof Proposition 2.19, they are defined by weighted initial forms of the series of the system S(Γ).


Definition 2.21. Assume that the rooted splice diagram Γr satisfies the semigroup condition and considera fixed strict splice type system S(Γ) associated to the (unrooted) splice diagram Γ. For each node v of Γ

and each index i ∈ {1, . . . , δv}, we let hv,i(z) ∈ C[zλ : λ ∈ ∂rΓ] be the polynomial obtained from fv,i(z) by

removing the term corresponding to the unique edge adjacent to v and pointing towards r. The subvarietyof C∂rΓ ≃ Cn−1 defined by the vanishing of (hv,i(z))v,i is called the end-curve of S(Γ) relative to r. We

denote it by Cr .

A planar embedding of Γr determines an ordering of the edges adjacent to a fixed node v and pointingaway from r (for example, by reading them from left to right). Once this order is fixed, by the Hamm

determinant conditions, each group of equations (hv,i(z) = 0)δv−2i=1 becomes equivalent to a collection of

wv-homogeneous binomial equations of degree ℓr,v of the form

zmv,ej − av,jzmv,eδv−1 = 0 for j ∈ {1, . . . , δv − 2},

with all av,j 6= 0. The next statement summarizes the main properties of Cr discussed in [32, Theorem 3.1]:

Theorem 2.22. The subvariety Cr ⊆ Cn−1 is a reduced complete intersection curve, smooth away from theorigin, and meets any coordinate subspace of Cn−1 only at the origin. It has g many components, whereg := gcd{ℓr,λ : λ ∈ ∂rΓ}, and all are isomorphic to torus-translates of the monomial curve in Cn−1 with

parameterization t 7→ (tℓr,λ1/g, . . . , tℓr,λn−1

/g).

Example 2.23. We fix the splice diagram from Example 2.16 and consider its rooted analog obtained bysetting the first leaf as its root r, as seen in the right of Figure 1. By construction, ℓr,2 = 49, ℓr,3 = 30,ℓr,4 = 42, ℓr,5 = 105, ℓr,u = 147, ℓr,v = 210, wu = (49, 30, 42, 105) and wv = (70, 10, 14, 35). The equationsdefining this end-curve are obtained by removing the monomial indexed by the edge pointing towards r ineach equation from (2.12). Since all weights are coprime, the curve Cr is reduced and irreducible. It isdefined as the solution set to

−2 z32 + z4 z5 = z73 + z54 − 2155 z25 = z73 + 2 z54 − 2123 z25 = 0.

Linear combinations of the last two expressions yield the equivalent binomial system:

−2 z32 + z4 z5 = z54 + 32z25 = z73 − 2187z25 = 0.

An explicit parameterization is given by (z2, z3, z4, z5) = (−t49, 3 t30,−2 t42, t105). ⋄

The collection hv,i(z) of polynomials defining the end-curve Cr determines a map Gr : Cn−1 → Cn−2. Ournext result, which we state for comparison’s sake with Corollary 7.9, discusses the restriction of this map toeach coordinate hyperplane of Cn−1:

Corollary 2.24. For every λ ∈ ∂rΓ, the restriction of Gr to the hyperplane Hλ of Cn−1 defined by theequation zλ = 0 is dominant.

Proof. By Theorem 2.22, the fiber over the origin of the restricted map Gr|Hλis finite. Upper semicontinuity

of fiber dimensions implies that the generic fiber is also 0-dimensional. Since dimHλ = n−2, the map Gr |Hλ

must be dominant. �

3. Local tropicalization

In [43], the last two authors developed a theory of local tropicalizations of algebraic, analytic or formalgerms endowed with maps to (not necessarily normal) toric varieties, adapting the original formulationof global tropicalization (see, e.g. [23]) to the local setting. In this section, we recall the basics on localtropicalizations that will be needed in Section 6. We focus our attention on germs (Y, 0) → Cn defined by

ideals I of the ring of convergent power series O := C{z1, . . . , zn} near the origin, rather than over its

completion O := CJz1, . . . , znK. As Remark 3.4 confirms, both local tropicalizations yield the same set.

The notion of local tropicalization of an embedded germ is rooted on the construction of initial idealsassociated to non-negative weight vectors, which we now describe. Any weight vector w := (w1, . . . , wn) ∈


(R≥0)n induces a real-valued valuation on O, known as the w-weight, as follows. Given a monomial zα :=

zα1

1 · · · zαnn , we set w(zα) := w · α =

∑ni=1 wi αi. In turn, for each f =

∑

α cα zα with f 6= 0 we set

(3.1) w(f) := min{w(zα) : cα 6= 0} = min{w · α : cα 6= 0}.

We define w(0) :=∞. The set {α : cα 6= 0} is called the support of f , and it is the basis of the constructionof the Newton polyhedron and Newton fan of f (see Definition 3.2).

Definition 3.1. Given f ∈ O and w ∈ (R≥0)n, the w-initial form inw(f) is the sum of the terms in the

series f with minimal w-weight w(f). In turn, given an ideal I of O, the w-initial ideal inw(I)O of I in

O is generated by the w-initial forms of all elements of I. If w ∈ (R>0)n, the w-initial ideal inw(I) of I is

the ideal of C[z1, . . . , zn] generated by the w-initial forms of all elements of I. If (Y, 0) → Cn is the germ

defined by I, the subscheme inw(Y ) → Cn defined by inw(I) is called the w-tangent cone of Y .

The initial forms of a series f ∈ O determine the Newton fan of f as follows:

Definition 3.2. The Newton polyhedron NP(f) of a non-zero element f ∈ O is the convex hull of the

Minkowski sum of the support of f and of (R≥0)n. Given a face K of NP(f), we let σK be the closure

of the set of weight-vectors w in (R≥0)n supporting K (that is, such that the convex hull of the support of

inw(f) is K). The set {σK : K face of NP(f)} is the Newton fan NF(f) of f .

The map K → σK yields an inclusion reversing bijection between the set of faces of NP(f) and theNewton fan NF(f). Furthermore, every face K of NP(f) satisfies:(3.2) dimK + dim σK = n.

Definition 3.3. Let (Y, 0) ⊆ Cn be a germ defined by an ideal I of O. The local tropicalization of I or of thegerm Y , is the set of all vectors w ∈ (R≥0)

n such that the w-initial ideal inw(I)O ⊆ O of I is monomial-free.

We denote it by Trop I or TropY . In turn, the positive local tropicalization of I or of the germ Y , is

the intersection of the local tropicalization with the positive orthant (R>0)n. We denote it by Trop>0 I or

Trop>0 Y .

Even though this definition depends heavily on the fixed embedding (Y, 0) ⊆ Cn, we omit it from the notationfor the sake of simplicity. The next remarks clarify some differences between the present approach and thatof [43], which defines local tropicalizations using local valuation spaces (see [43, Definitions 5.13 and 6.7]). Arecent extension of this construction to toric prevarieties by means of Berkovich analytification can be foundin [21].

Remark 3.4. As shown in [43, Theorem 11.2], [48, Corollary 4.3], local tropicalization (involving either O or

O) admit several equivalent characterizations analogous to the Fundamental Theorem of Tropical AlgebraicGeometry [23, Theorem 3.2.3]. One of them is as Euclidean closures in (R≥0)

n of images of local valuation

spaces. By [43, Corollary 5.17], the canonical inclusion (O,m) → (O, m) induces an isomorphism of local

valuation spaces. This implies that extending an ideal in O to the complete ring O will yield the same localtropicalization. Therefore, we can define local tropicalizations for ideals of O rather than of O, in agreementwith the setting of splice type systems.

Remark 3.5. Our choice of terminology for local tropicalizations differs slightly from [43], as we now explain.As was shown in [43, Section 6], the local tropicalizations of the intersection of a germ (Y, 0) → Cn witheach coordinate subspace of Cn can be glued together to form an extended fan in (R≥0 ∪ {∞})n calledthe local nonnegative tropicalization of Y in [43]. In the present paper, we refer to this structure as anextended tropicalization of the germ Y , in agreement with Kashiwara and Payne’s constructions for globaltropicalizations (see [23, §6.2]). The finite part of this extended tropicalization (i.e., its intersection with(R≥0)

n) is the local tropicalization from Definition 3.3. A precise description of the boundary strata of theextended local tropicalization of splice type singularities is given in Subsection 6.2.


Remark 3.6. The local tropicalization of a germ (Y, 0) ⊆ Cn coincides with the local tropicalization of the

associated reduced germ (Yred, 0) ⊆ Cn since the defining ideal of (Yred, 0) is the radical√I ⊆ O and w-

initial forms respect products. As a consequence, inw(√I) is monomial-free if and only if the same is true for

inw(I). Alternatively, the same statement can be obtained from the definition of local tropicalization as animage of the local valuation space of Y and the fact that the embedding Yred → Y induces a homeomorphismof local valuation spaces (see [43, Lemma 5.18]).

Remark 3.7. The local tropicalization of a reduced germ (Y, 0) ⊆ Cn is equal to the union of local tropical-izations of its irreducible components, and the same is true for their local positive tropicalizations. This isa direct consequence of the fact that the local valuation space of (Y, 0) is the union of the local valuationspaces of its irreducible components (see [43, Lemma 5.18]).

The next result determines the local tropicalization of a germ (Y, 0) → Cn from the positive one:

Proposition 3.8. The local tropicalization TropY is the closure of the positive local tropicalization Trop>0 Yinside the cone (R≥0)

n.

Proof. By Remarks 3.6 and 3.7, it suffices to consider the case where (Y, 0) is irreducible. In this situation, [43,Thm. 11.9] shows that the extended local tropicalization of (Y, 0) is the closure of the extended positive localtropicalization in the extended non-negative orthant (R≥0 ∪ {∞})n. The result then follows by intersectingthe extended local tropicalization with (R≥0)

n. �

As in the global case, local tropicalizations of hypersurface germs can be obtained from the correspondingNewton fans. Indeed, if the ideal I is principal, generated by f ∈ O, and w ∈ (R≥0)

n, then each initial idealinw(I)O is also principal, with generator inw(f). Equality (3.2) then yields the following statement (see [43,Proposition 11.8]):

Proposition 3.9. The set Trop(f) is the union of all codimension-one cones of the Newton fan of f .

By contrast, if I has two or more generators {f1, . . . , fs}, their w-initial forms need not generate inw(I)O.However, for the purpose of characterizing Trop>0 I, it is enough to have a tropical basis for I in the senseof [43, Definition 10.1], i.e., a finite set of generators {f1, . . . , fs} of I which is a universal standard basis ofI in the sense of [43, Definition 9.8] and such that for any w ∈ (R>0)

n, the w-initial ideal inw(I)O containsa monomial if and only if one of the initial forms inw(fi) is a scalar multiple of a monomial.

Remark 3.10. Such tropical bases exist by [43, Theorem 10.3] and can be used to determine Trop I byintersecting the local tropicalizations of the corresponding hypersurface germs. Furthermore, their existenceensures that local tropicalizations are supports of rational fans in (R≥0)

n. Indeed, the corresponding fanis obtained by considering the common refinement of the intersection of the local tropicalization of eachmember of a tropical basis for I combined with Proposition 3.9. Furthermore, under the hypothesis that noirreducible component of (Y, 0) is included in a coordinate subspace of Cn, such fans and their refinementsare standard tropicalizing fans of Y in the sense of Definition 3.13 below.

In what follows, we restrict our attention to subgerms of (Cn, 0) with no components included in coordinatesubspaces. Their local tropicalizations verify the following key property (see [43, Proposition 9.21, Theorems10.3 and 11.9]):

Proposition 3.11. Let (Y, 0) be a subgerm of (Cn, 0) with no irreducible component contained in a coordinatesubspace of Cn and let I ⊆ O be its defining ideal. Then, TropY is the support of a rational polyhedral fanF which satisfies the following conditions:

(1) the dimension of all the maximal cones of F agrees with the complex dimension of Y ;(2) the maximal cones of F have non-empty intersections with (R>0)

m;(3) given any cone τ of F , the w-initial ideal of I is independent of the choice of w ∈ τ◦.

Proof. By Remarks 3.6 and 3.7, we may restrict to the case when (Y, 0) is reduced and irreducible. Theexistence of F satisfying the first two conditions is a direct consequence of [43, Theorem 11.9], which ensuresthe analogous properties hold for the extended local tropicalization of (Y, 0). Note that TropY is non-empty,since (Y, 0) meets the dense torus of Cn (see [43, Lemma 7.4]). Furthermore, the fan structure on TropYinduced from a tropical basis {f1, . . . , fr} for I discussed in Remark 3.10 satisfies condition (3). In turn, anyrefinement will satisfy conditions (1) and (2) as well. �


Remark 3.12. Proposition 3.11 allows us to recover the complex dimension of an irreducible germ (Y, 0) → Cn

meeting the dense torus from its positive tropicalization (see, for instance, the proofs of Corollaries 6.18and 6.19). Indeed, its dimension agrees with the dimension of any of the top-dimensional cones in anyfixed fan satisfying conditions (1)–(3) in Proposition 3.11. If (Y, 0) is not irreducible, a similar proceduredetermines the maximal dimension of a component of the germ meeting the dense torus.

Definition 3.13. Let (Y, 0) be a subgerm of Cn with no irreducible component contained in a coordi-nate subspace of Cn. Let I be the ideal of O defining (Y, 0). Any fan F satisfying all three conditionsin Proposition 3.11 is called a standard tropicalizing fan for (Y, 0) or for I. For every cone τ of F meeting

(R>0)n, we write inτ (I) := inw(I) and inτ (Y ) := inw(Y ), for any w ∈ τ◦ (see Definition 3.1).

Remark 3.14. When Y is a hypersurface germ defined by a series f ∈ O, the set of cones of codimension oneof the Newton fan of f satisfies all three conditions listed in Proposition 3.11. Furthermore, it is the coarsestfan with these properties. However, for germs of higher codimension such canonical choice need not alwaysexist. For an example in the global (i.e., polynomial) setting, we refer the reader to [23, Example 3.5.4].

The next proposition emphasizes the relevance of tropicalizing fans for producing birational models ofirreducible germs with desirable geometric properties, in the spirit of Tevelev’s construction of tropicalcompactifications of subvarieties of tori [50].

Proposition 3.15. Fix a rational polyhedral fan F whose cones are contained in (R≥0)n and let πF : XF →

Cn be the associated toric morphism. Given an irreducible germ (Y, 0) → Cn meeting the dense torus, letYF be the strict transform of Y under πF and write π : YF → Y for the restriction of the map πF to YF .Then, the following properties hold:

(1) The restriction π is proper if and only if the support |F| contains the local tropicalization TropY .(2) Assume that π is proper. Then, the strict transform YF intersects every orbit S of XF along a non-

empty pure-dimensional subvariety with codimY (YF ∩S) = codimXF(S) if and only if |F| = TropY .

Proof. In what follows, we use standard terminology and notation from toric geometry, which can be foundin Fulton’s book [9]. The proof of (2) is similar to the global analog [23, Proposition 6.4.7 (2)], so we leaveit to the reader.

It remains to prove assertion (1). To this end, we consider a fan Σ subdividing the non-negative cone(R≥0)

n containing F as a subfan. Such a fan exists by [7, Theorem 2.8 (III.2)].We consider the toric varieties XF and XΣ associated to the fans F and Σ and the natural toric morphisms

πF : XF → Cn and πΣ : XΣ → Cn. We let YΣ and YF be the strict transforms of Y under these two maps.The aforementioned varieties and maps fit naturally into the commutative diagram

YΣ� � // XΣ

πΣ

!!❈❈

❈❈❈❈

❈❈XF? _oo

πF}}④④④④④④④④

YF? _oo

π}}④④④④④④④④

Cn Y?_oo

where the central triangle involves toric morphisms and the horizontal arrows are embeddings. The verticalmap π on the right is the restriction of πF to YF . In what follows we view XF as an open subvariety of XΣ.Note that the toric birational morphism πΣ is proper by [9, Section 2.4], as the defining fans of its sourceand target have the same support, namely (R≥0)

n.By construction, π is proper if and only if YΣ is contained in XF . Thus, claim (1) will follow if we show

that for every cone τ of Σ we have the following equivalence:

(3.3) Oτ ∩ YΣ 6= ∅ ⇐⇒ τ◦ ∩ TropY 6= ∅,where Oτ := Hommonoid(τ

⊥ ∩M,C) is the corresponding toric orbit.It remains to prove (3.3). We start with the forward implication and fix y0 ∈ Oτ ∩YΣ. Then, there exists

a holomorphic arc in Y ∩ (C∗)n parameterized as t 7→ y(t) such that the limit as t→ 0 of its strict transformin YΣ equals y0, i.e.,

limt→0

y(t) = y0 ∈ Oτ ∩ YΣ.

Such an arc can be built by choosing an irreducible subgerm of a curve CΣ of the germ (YΣ, y0), notcontained in the toric boundary, then by projecting it to a subgerm C of Y via πΣ and, finally, by choosinga normalization of C, which we identify with (C, 0).


We consider the weight vector w := ord y(t) in the dual lattice N := M∨ recording the orders of vanishingof the components of y(t). We claim that w belongs to τ◦ ∩ TropY , so τ◦ ∩ TropY 6= ∅, as desired. Toprove this claim, notice that the arc t 7→ yin(t) of YΣ obtained by keeping the w-initial terms yin(t) of thecomponents of y(t) has the same limit when t→ 0 as y(t) does. This fact can be checked by working in theaffine toric variety associated to the cone R≥0〈w〉. Properties of limits in toric varieties from [9, Section 2.3]ensure that limt→0 yin(t) ∈ Oτ is equivalent to the fact that the weight vector w lies in τ◦ ∩N . Therefore,we conclude that ord y(t) ∈ τ◦.

It remains to verify that w ∈ TropY . To do so, it suffices to notice that TropC ⊆ TropY (since C ⊆ Y )and that TropC = R≥0〈w〉. The latter is a direct consequence of [48, Corollary 4.3 3’)] (see also Maurer’spaper [24] which includes a precursor of local tropicalization for germs of space curves).

Finally, to prove the reverse implication of (3.3), we assume that τ◦∩TropY 6= ∅ and let w ∈ τ◦∩TropY bea primitive lattice vector in N ≃ Zn. We consider a refinement Σw of Σ such that the ray τw = R≥0〈w〉 ∈ Σw .By construction, the orbit Oτw is mapped via the toric morphism πw : XΣw

→ XΣ to the orbit Oτ .The intersection of YΣw

with the orbit Oτw is determined by the w-initial ideal inw(I(Y )) of the idealI(Y ) defining Y . Since this initial ideal is monomial free because w ∈ TropY , this intersection is nonempty.Since Oτw ⊂ YΣw

, the map πw ensures that YΣ ∩ Oτ 6= ∅ as well. This concludes our proof. �

One well-known feature of global tropicalizations of equidimensional subvarieties of toric varieties is theso-called balancing condition [23, Section 3.3]. To this end, tropical varieties must be endowed with positiveinteger weights (tropical multiplicities) along their top-dimensional cones. Such multiplicities may be alsodefined in the local situation. We restrict the exposition to irreducible germs, since this is sufficient for thepurposes of this paper.

Definition 3.16. Let (Y, 0) → Cn be an irreducible germ meeting (C∗)n, defined by an ideal I of O andlet F be a standard tropicalizing fan for it. Given a top-dimensional cone τ of F , we define the tropicalmultiplicity of F at τ to be the number of irreducible components of inτ (Y )∩(C∗)n, counted with multiplicity.

Remark 3.17. Balancing for local tropicalizations of equidimensional germs follows from [43, Remark 11.3,Theorem 12.10] and features in the proof of Proposition 6.24. This property will help us prove that theembedded splice diagrams in Rn are included in the local tropicalizations of the corresponding splice typesystems (see Subsection 6.3). Tropical multiplicities will be also used in Section 8 to recover the edge weightson any coprime splice diagram Γ from the local tropicalization of any splice type surface singularity associatedto it.

4. Newton non-degeneracy

In this section we discuss the notion of Newton non-degenerate complete intersection in the sense ofKhovanskii [17], starting with the case of formal power series in O, as introduced by Kouchnirenko in [19,Sect. 8] and [20, Def. 1.19]. Kouchnirenko’s definition was later extended by Steenbrink [47, Def. 5] toC-algebras of formal power series CJP K with exponents on an arbitrary saturated sharp toric monoid P . Forprecursors to this notion we refer the reader to Teissier’s work [49, Section 5].

Definition 4.1. Given a series f ∈ O, we say that f is Newton non-degenerate if for any positive weightvector w ∈ (R>0)

n, the subvariety of the dense torus (C∗)n defined by inw(f) is smooth.

The notion of Newton non-degeneracy extends naturally to finite sequences of functions. For our purposes,it suffices to restrict ourselves to regular sequences, i.e., collections (f1, . . . , fs) in O where

(1) f1 is not a zero divisor of O, and(2) for each i ∈ {1, . . . , s−1}, the element fi+1 is not a zero divisor in the quotient ring O/〈f1, . . . , fi〉O.

As O is a regular local ring, the germs defined by regular sequences in O are exactly formal completeintersections at the origin of Cn.

Definition 4.2. Fix a positive integer s and a regular sequence (f1, . . . , fs) in O. Consider the germ

(Y, 0) → Cn defined by the ideal (f1, . . . , fs)O. The sequence (f1, . . . , fs) is a Newton non-degeneratecomplete intersection system for (Y, 0) if for any positive weight vector w ∈ (R>0)

n, the hypersurfacesof (C∗)n defined by each inw(fi) form a normal crossings divisor in a neighborhood of their intersection.Equivalently, the differentials of the initial forms inw(f1), . . . , inw(fs) must be linearly independent at eachpoint of this intersection.


Remark 4.3. Notice that Definition 4.2 allows for the initial forms to be monomials. This would determinean empty intersection with the dense torus (C∗)n.

Remark 4.4. Definition 4.2 modifies slightly Khovanskii’s original definition from [17, Rem. 4 of Sect. 2.4],by imposing the regularity of the sequence (f1, . . . , fs). The generalization to formal power series CJP Kassociated to an arbitrary sharp toric monoid P is straightforward, and parallels that done by Steenbrink [47]for hypersurfaces. A slightly different notion of Newton non-degenerate ideals was introduced by Saia in [44]

for ideals of finite codimension in O. For a comparison with Khovanskii’s approach we refer the readerto Bivia-Ausina’s work [4, Lemma 6.8]. For a general perspective on Newton non-degenerate completeintersections, the reader can consult Oka’s book [35]. A definition of Newton non-degenerate algebraicsubgerms of (Cn, 0) for not necessarily complete intersections was given by Aroca, Gomez-Morales andShabbir [1], where I ⊂ O is Newton non-degenerate if for every weight vector w with positive entries,the initial ideal inw(I) defines a smooth subscheme of the torus (C∗)n. The results of their paper (andthe proofs involving Grobner bases) can be extended to the analytic and formal contexts by working withstandard bases, in the spirit of [43]. Recent work of Aroca, Gomez-Morales and Mourtada [2] generalize theconstructions from [1] to subgerms of arbitrary normal affine toric varieties.

5. Embeddings and convexity properties of splice diagrams

In this section, we describe simplicial fans in the real weight space N(∂ Γ) ⊗Z R ≃ Rn that arise fromsplice diagrams and appropriate subdiagrams. These constructions will play a central role in Section 6,when characterizing local tropicalizations of splice type systems. Throughout this section we assume thatthe splice diagram Γ satisfies the edge determinant condition of Definition 2.8. The semigroup conditionfrom Definition 2.12 plays no role.

We let ∆n−1 be the standard (n− 1)-simplex in Rn with vertices {wλ : λ ∈ ∂ Γ}. We start by defining

a piecewise linear map from Γ to ∆n−1:

(5.1) ι : Γ→ ∆n−1 where ι(v) =wv

|wv|for each vertex v of Γ.

Here, |·| denotes the 1-norm in Rn. In particular, |wv| =∑

λ∈∂ Γ ℓv,λ for each node v of Γ. After identifyingeach edge e of Γ with the interval [0, 1], the map ι on e is defined by convex combinations of the assignmentat its endpoints. The injectivity of ι will be discussed in Theorem 5.11.

The following combinatorial constructions, in particular Definitions 5.1, 5.3 and 5.4, play a prominentrole in proving Theorem 6.2. Stars of vertices described in Definition 2.1 and convex hulls of vertices(see Definition 2.2) are central to many arguments below.

Definition 5.1. A subtree T of the splice diagram Γ is star-full if StarΓ(v) ⊂ T for every node v of T . Anode of T is called an end-node if it is adjacent to exactly one node of T .

Every tree that is not a star tree contains at least two end-nodes. The following statement, illustratedin Figure 2, describes a method to produce new star-full subtrees from old ones by pruning from an end-node.Its proof is straightforward, so we omit it.

Lemma 5.2. Let T be a star-full subtree of Γ with d leaves {u1, . . . , ud}. Fix an end-node v of T and assumethat u1, . . . , uδv−1 are the only leaves of T adjacent to v. Then, their convex hull T ′ := [v, uδv , . . . , ud] is alsostar-full.

Definition 5.3. A branch of a tree T adjacent to a node v is a connected component of

T r({v} ∪

⋃

e∈StarΓ(v)

e◦),

where e◦ denotes the interior of the edge e.

For example, {u1}, . . . , {us} and the tree T ′ := [v′, T1, . . . Ts′ ] on the left of Figure 2 are the δv branches ofT adjacent to the node v. Similarly, the branches of T ′ adjacent to v′ are T1, . . . , Ts′ and {v}.

Star-full subtrees of splice diagrams have a key convexity property rooted in barycentric calculus. This isthe content of Proposition 5.10. The following definition plays a central role in its proof.


Figure 2. Pruning a star-full subtree T of Γ from an end-node v of T produces a newstar-full subtree T ′ with one fewer node, as in Lemma 5.2. Here, s = δv−1 and s′ = δv′ −1.

Definition 5.4. Let v be a node of Γ and let L be a collection of leaves of Γ. We define Bar(L; v) as thebarycenter of the leaves in L with weights determined by wv, that is:

(5.2) Bar(L; v) :=∑

λ∈L

ℓv,λℓ

wλ where ℓ :=∑

λ∈L

ℓv,λ.

In particular, Bar({λ}; v) = wλ for any leaf λ.

Remark 5.5. Fix a node v of Γ with adjacent branches Γ1, . . . ,Γδv . Then, the set {Bar(∂ Γi; v) : i = 1, . . . , δv}is linearly independent, and a direct computation gives

(5.3) ι(v) =

δv∑

j=1

( 1

|wv|∑

λ∈∂ Γj

ℓv,λ)Bar(∂ Γj ; v).

In particular, ι(v) lies in the relative interior of the simplex conv({Bar(∂ Γj ; v) : j = 1, . . . , δv})), where conv

denotes the affine convex hull inside ∆n−1.

Following the notation from Definition 2.12, we write ∂aΓ := ∂b,[a,b]Γ and ∂bΓ := ∂a,[a,b]Γ for each pair

of adjacent nodes a, b of Γ. Barycenters determined by splitting Γ along the edge [a, b] are closely related,as the following lemma shows:

Lemma 5.6. Let a, b be two adjacent nodes of Γ, with associated sets of leaves ∂aΓ and ∂bΓ on each side ofΓ. Then:

(1) Bar(∂aΓ; a) = Bar(∂aΓ; b) and Bar(∂bΓ; a) = Bar(∂bΓ; b) (which we denote by Bar(∂aΓ; [a, b]) andBar(∂bΓ; [a, b]), respectively).

(2) The points ι(a) and ι(b) lie in the line segment [Bar(∂aΓ; [a, b]),Bar(∂bΓ; [a, b])].(3) Bar(∂aΓ; [a, b]) < ι(a) < ι(b) < Bar(∂bΓ; [a, b]), where < is the order given by identifying the segment

[Bar(∂aΓ; [a, b]),Bar(∂bΓ; [a, b])] with [0, 1].

Proof. We start by showing (1). Definition 2.5 induces the following identities:

(5.4) ℓb,λ = ℓa,λℓa,bda

for all λ ∈ ∂aΓ, and ℓa,µ = ℓb,µℓa,bdb

for all µ ∈ ∂bΓ.

Thus, a and b contribute proportional weights to ∂aΓ and ∂bΓ, respectively. This implies that the corre-sponding barycenters agree, proving (1).

Next, we discuss (2). To simplify notation, we write w = Bar(∂aΓ; [a, b]) and w′ = Bar(∂bΓ; [a, b]).Then, (5.4) yields:

wa =( ∑

λ∈∂aΓ

ℓa,λ)w +

( ∑

µ∈∂bΓ

ℓa,µ)w′ and wb =

( ∑

λ∈∂aΓ

ℓb,λ)w +

( ∑

µ∈∂bΓ

ℓb,µ)w′.

The definition of ι then confirms that ι(a) and ι(b) are convex combinations of w and w′.It remains to prove (3). The condition w < ι(a) < ι(b) < w′ claimed in (3) is equivalent to:

( ∑

λ∈∂aΓ

ℓa,λ) ( ∑

µ∈∂bΓ

ℓa,µ)−1

>( ∑

λ∈∂aΓ

ℓb,λ) ( ∑

µ∈∂bΓ

ℓb,µ)−1

.


Rearranging these expressions by sums with common indexing sets and simplifying further using (5.4) yieldsthe equivalent identity:

(5.5) 1 >( ∑

λ∈∂aΓ

ℓb,λ) ( ∑

λ∈∂aΓ

ℓa,λ)−1 ( ∑

µ∈∂bΓ

ℓa,µ)( ∑

µ∈∂bΓ

ℓb,µ)−1

=ℓa,bda

ℓa,bdb

.

The edge determinant condition for [a, b] confirms the validity of (5.5), and so (3) holds. �

Our next result shows that the image under ι of the vertices adjacent to a fixed node v satisfy a convexityproperty analogous to that of Remark 5.5:

Proposition 5.7. Let v be a node of Γ, with adjacent vertices u1, . . . , uδv . Then:

(1) {ι(ui) : i = 1, . . . , δv} is linearly independent;

(2) ι(v) ∈(conv({ι(uj) : j = 1, . . . , δv})

)◦.

Proof. Both statements are clear if v is only adjacent to leaves of Γ (i.e., when Γ is a star splice diagram), sowe may assume v is adjacent to some node of Γ. Thus, up to relabeling if necessary, we suppose {u1, . . . , us}are leaves of Γ and {us+1, . . . , uδv} are nodes of Γ (we set s = 0 if v is only adjacent to nodes).

For each j ∈ {1, . . . , δv}, we let Γj be the branch of Γ adjacent to v containing uj , and set wj :=Bar(∂ Γj ; v). Lemma 5.6 (2) and the definition of barycenters ensures the existence of αj ∈ (0, 1] satisfying

(5.6) ι(uj) = αjwj + (1 − αj) ι(v) for each j ∈ {1, . . . , δv},with αj = 1 if and only if j ∈ {1, . . . , s}.

We start by proving (1). We fix a linear relation∑δv

j=1 βj ι(uj) = 0. Substituting (5.6) into this depen-dency relation yields

(5.7)

δv∑

j=1

(βj αj)wj +B ι(v) = 0 where B :=

δv∑

j=1

βj (1 − αj).

We claim that B = 0. Indeed, assuming this is not the case, we use (5.7) to write ι(v) in terms of w1, . . . , wδv .Comparing this expression with (5.3) and using the linear independence of {w1, . . . , wδv} gives:

(5.8) βj αj = −B

|wv|∑

λ∈∂ Tj

ℓv,λ for each j ∈ {1, . . . , δv}.

In particular, all βi are non-zero and have the same sign, namely the opposite sign to B. Summing up the

expressions (5.8) over all j yields∑δv

j=1 βj = 0, which cannot happen due to the sign constraint on the βj ’s.From this it follows that B = 0.

Since B = 0, the linear independence of {w1, . . . , wδv} forces βjαj = 0 for all j. Combining this with ourassumption that αj > 0 for all j, gives βj = 0 for all j = 1, . . . , δv, thus confirms (1).

To finish, we discuss (2). We let q1, . . . , qδv be the coefficients used in (5.3) to write ι(v) as a convexcombination of w1, . . . , wδv . Substituting the value of each wj obtained from (5.6) in expression (5.3) yields:

(5.9) ι(v) =

s∑

j=1

qjA

ι(uj) +

δv∑

j=s+1

qjAαj

ι(uj) where A := 1 +

δv∑

j=s+1

qj (1− αj)

αj.

The conditions 0 < αj < 1 for j ∈ {s+1, . . . , δv}, and the definition of q1, . . . , qδv ensure that the right-handside of (5.9) is a positive convex combination of ι(u1), . . . , ι(uδv ), as we wanted to show. �

Each subtree T of Γ determines a polytope via the map ι:

(5.10) ∆T := conv({ι(u) : u ∈ ∂ T }) ⊆ ∆n−1 .

For example, ∆Γ is the standard simplex ∆n−1. We view the next result as a key convexity property ofstar-full subtrees of splice diagrams.

Lemma 5.8. Fix a star-full subtree T of Γ. For every node u of T , ι(u) admits an expression of the form

(5.11) ι(u) =∑

µ∈∂ T

αµ ι(µ) with∑

µ∈∂ T

αµ = 1 and αµ > 0 for all µ ∈ ∂ T.

In particular, ι(T ) ⊂ ∆T .


Proof. We let p be the number of nodes of T and proceed by induction on p. The statement is vacuous forp = 0. If p = 1, then T is a star tree and the result follows by Proposition 5.7. For the inductive step, welet p ≥ 2 and suppose that the result holds for star-full subtrees with (p− 1) nodes.

We fix ∂ T = {u1, . . . , ud} and we let v be an end-node of T . Following Figure 2, we let v′ be the uniquenode of T adjacent to v and assume that u1, . . . , uδv−1 are adjacent to v. Proposition 5.7 (2) applied toStarΓ(v) gives

(5.12) ι(v) = β0 ι(v′) +

δv−1∑

j=1

βj ι(uj) with

δv−1∑

j=0

βj = 1 and βj > 0 for all j.

Using Lemma 5.2, we let T ′ be the star-full subtree of Γ obtained by pruning T from v. By construction,T ′ has (p − 1) nodes and its leaves are {v} ∪ {uδv , . . . , ud}. The inductive hypothesis yields the followingexpressions for v′ and all other (potential) nodes u 6= v′ of T ′:

(5.13) ι(v′) = γ0 ι(v) +

d∑

j=δv

γj ι(uj) and ι(u) = α0 ι(v) +

d∑

j=δv

αj ι(uj),

with α0 +

d∑

j=δv

αi = γ0 +

d∑

j=δv

γj = 1 and αi, γi > 0 for all i ∈ {0, δv, . . . , d}. Since 0 < γ0β0 < 1, substituting

the expression for ι(v′) obtained from (5.13) into (5.12) produces the desired positive convex combinationfor ι(v):

ι(v) =

δv−1∑

j=1

βj

1− γ0β0ι(uj) +

d∑

j=δv

β0γj1− γ0β0

ι(uj).

In turn, substituting this identity in both expressions from (5.13) gives the positive convex combinationstatement for all remaining nodes of T . The inclusion ι(T ) ⊂ ∆T follows by the convexity of ∆T . �

Proposition 5.7 and Lemma 5.8 combined have the following natural consequence:

Corollary 5.9. For each pair of star-full subtrees T, T ′ of Γ with T ′ ⊆ T , we have ∆T ′ ⊆ ∆T .

Our next result is a generalization of Proposition 5.7 and it highlights a key combinatorial property sharedby Γ and all its star-full subtrees.

Proposition 5.10. Let T be a star-full subtree of Γ. Then:

(1) the weights {ι(u) : u ∈ ∂ T } are linearly independent;(2) ∆T is a simplex of dimension |∂ T | − 1;(3) for each node v of T we have ι(v) ∈ ∆T

◦.

Proof. Item (2) is a direct consequence of (1). In turn, item (3) follows from (2) and Lemma 5.8. Thus, itremains to prove (1). We distinguish two cases, depending on the number of nodes of T , denoted by p.

Case 1: If p < 2, then T is either a vertex, an edge of Γ, or a star tree. If T is a vertex of Γ, then the claimholds because ι(u) 6= 0 for any vertex u of Γ. If T is a star tree, the statement agrees with Proposition 5.7 (1).

Next, assume T is an edge of Γ. We consider two scenarios. First, if T joins a leaf λ and a node u of Γ,then the result follows immediately since ι(u) and wλ are linearly independent. On the contrary, assume Tjoins two adjacent nodes of Γ, say u and v. Pick two leaves λ, µ of Γ with u ∈ [λ, v] and v ∈ [u, µ] (i.e., λis on the u-side and µ is on the v-side of Γ, as seen from the edge [u, v]). Lemma 2.7 yields the followingformula for the (λ, µ)-minor of the matrix (wu|wv):

(ℓλ,u ℓλ,vℓµ,u ℓµ,v

)

= ℓλ,u ℓµ,vdu,vdv,u − ℓu,v

du,vdv,u=

ℓλ,u ℓµ,v det([u, v])

du,vdv,u.

This expression is positive by the edge determinant condition, so the set {ι(u), ι(v)} is linearly independent.Case 2: If p ≥ 2, we know that d := |∂ T | ≥ 4. We prove the result by reverse induction on d. When

d = n, we have T = Γ and there is nothing to prove since ι(u) = wu for all u ∈ ∂ Γ. Next, we fixd ∈ {4, . . . , n− 1} and let T be a star-full tree with d leaves, labeled u1, u2, . . . , ud. Assume that the linearindependence holds for any star-full subtree with k ≥ d+ 1 leaves. Without loss of generality we assume u1

is a node of Γ (one must exist since T 6= Γ is star-full). Set s = δu1− 1.


Next, we define T ′ := T ∪StarΓ(u1). By construction, T ′ is a star-full subtree of Γ with (d+ s− 1) leaves.Since u1 is a node of T ′, Lemma 5.8 applied to T ′ yields a positive convex combination:

(5.14) ι(u1) =∑

v∈∂ T ′

βv ι(v) with∑

v∈∂ T ′

βv = 1 and βv > 0 for all v ∈ ∂ T ′.

To prove the linear independence for the d points ι(u1), . . . , ι(ud), we fix a potential dependency relation∑d

j=1 αj ι(uj) = 0. Substituting (5.14) into it gives a linear dependency relation for the leaves of T ′:

d∑

j=2

(αj + α1 βuj) ι(uj) +

d+s∑

j=d+1

(α1 βuj) ι(uj) = 0,

where {ud+1, . . . , ud+s} are the leaves of T ′ adjacent to u1. The inductive hypothesis applied to T ′ and thepositivity of each βv with v ∈ ∂ T ′ forces αj = 0 for all j = 1, . . . , d. Thus, (1) holds. �

Next, we state the main result in this section, which is a natural consequence of Lemma 5.8:

Theorem 5.11. The map ι from (5.1) is injective.

Proof. We prove the statement holds when restricted to any star-full subtree T of Γ. As in the proofof Lemma 5.8, we argue by induction on the number p of nodes of T . If p = 0, then T is either a vertexu or an edge [u, v]. The statement in the first case is tautological. The result for the second one holds byconstruction because {ι(u), ι(v)} is linearly independent by Proposition 5.7 (1).

If p = 1, then T is a star tree. Let v be its unique node and {u1, . . . , uδv} be its leaves. Injectivity overT is a direct consequence of the following identity:

ι([ui, v]) ∩ ι([uj, v]) = {ι(v)} for all i 6= j,

which we prove by a direct computation. Indeed, pick 0 ≤ a ≤ b ≤ 1 with

(5.15) a ι(ui) + (1 − a) ι(v) = b ι(uj) + (1− b) ι(v).

By Proposition 5.7 (2), ι(v) admits a unique expression:

ι(v) =

δv∑

k=1

αk ι(uk) with

δv∑

k=1

αk = 1 and αk > 0 for all k.

Substituting this identity in (5.15) yields the following affine dependency relation for ι(∂ T ):

(a+ (b− a)αi) ι(ui) + ((b − a)αj − b) ι(uj) +∑

k 6=i,j

αk(b− a) ι(uk) = 0

By Proposition 5.7 (1), we conclude that a+(b− a)αi = (b− a)αj − b = 0 and αk(b− a) = 0 for all k 6= i, j.Since αk > 0 for all k 6= i, j and δv ≥ 3, it follows that b−a = 0 and a = −b = 0. Therefore, expression (5.15)represents ι(v).

Finally, pick p ≥ 2 and assume the result holds for star-full subtrees with p− 1 nodes. Let T be a star-fullsubtree with p nodes and pick an end-node v of T . As in Figure 2, write v′ for the unique node of T adjacentto it and {u1, . . . , uδv−1} for the leaves of T adjacent to v.

As in Lemma 5.2, let T ′ be the star-full subtree obtained by pruning T from v. Our inductive hypothesisensures that ι is injective when restricted to T ′. By the p = 1 case we know that ι([v, ui])∩ ι([v, uj ]) = {ι(v)}if i 6= j. Thus, the injectivity of ι when restricted to T will be proven if we show:

(5.16) ι([v, ui]) ∩ ι(T ′) = {ι(v)} for all i = 1, . . . , δv − 1.

The identity follows from Proposition 5.10. Indeed, we write any w on the left-hand side of (5.16) as

(5.17) w := a ι(ui) + (1 − a) ι(v) ∈ ι([v, ui]) ∩ ι(T ′) with 0 ≤ a ≤ 1.

Recall thatw ∈ ι(T ′) ⊂ ∆T ′ and ∆T ′ ⊂ ∆T by Corollary 5.9. Since ι(v) ∈ (∆T )◦as in Proposition 5.10 (3),

substituting this expression into (5.17) and comparing it with the known expression for w as an element of∆T yields an affine dependency equation for {ι(u) : u ∈ ∂ T }. The positivity constraint on the coefficientsused to write ι(v) as an element of (∆T )

◦forces a = 0, and so (5.16) holds. �


6. Local tropicalization of splice type systems

Let Γ be a splice diagram with n leaves and let S(Γ) be a splice type system associated to it, asin Definition 2.13. Fixing a total order on ∂ Γ yields an embedding of the corresponding splice type singular-ity (X, 0) into Cn. In this section we describe the local tropicalization of this embedded germ by computingthe local tropicalization of the intersections of X with each coordinate subspace of Cn (see Definition 3.3).As a byproduct, we confirm the first half of Theorem 1.1, namely that (X, 0) is a complete intersection inCn with no irreducible components contained in any coordinate subspace.

The injectivity of the map ι from (5.1), discussed in Theorem 5.11, fixes a natural simplicial fan structureon the cone over ι(Γ) in Rn:

Definition 6.1. Let Γ be a splice diagram. Then, the set R≥0 ι(Γ) has a natural fan structure, withtop-dimensional cones

{R≥0 ι([u, v]) : [u, v] is an edge of Γ}.We call it the splice fan of Γ.

Here is the main result of this section:

Theorem 6.2. The local tropicalization of (X, 0) → Cn is supported on the splice fan of Γ.

We prove Theorem 6.2 in two steps by a double inclusion argument, first restricting our attention to thepositive tropicalization. In Subsection 6.1 we show that the positive local tropicalization of S(Γ) is containedin the support of the splice fan of Γ. We prove this fact by working with simplices associated to star-fullsubtrees of Γ, which were introduced in Definition 5.1. For clarity of exposition, we break the arguments intoa series of combinatorial lemmas and propositions. These results allow us to certify that the ideal generatedby the w-initial forms of all the series Fv,i in S(Γ) where w lies in the complement of splice fan of Γ in(R>0)

n, always contains a monomial.In turn, showing that the support of the splice fan of Γ lies in the Euclidean closure of the local tropical-

ization of (X, 0) involves the so-called balancing condition for pure-dimensional local tropicalizations. Thisis the subject of Subsection 6.3. An alternative proof will be given in Section 7 after proving the Newtonnon-degeneracy of the germ (X, 0).

The fact that the positive tropicalization of (X, 0) is pure-dimensional is established in an indirect way.Our proof technique relies on the explicit computation of the boundary components of the extended trop-icalization, which is done in Subsection 6.2. This establishes the first half of Theorem 1.1 discussed above(see Corollary 6.18). As a consequence, we confirm by Corollary 6.20 that the local tropicalization is theEuclidean closure of the positive one. This result together with the findings in Sections 6.1 and 6.3 completethe proof of Theorem 6.2.

Remark 6.3. Throughout the next subsections, we adopt the following convenient notation for the admissibleexponent vectors mv,e from (2.6). Given a node v and a vertex u of Γ with u 6= v, we define mv,u := mv,e

where e is the unique edge adjacent to v and lying in the geodesic [v, u]. Similarly, given a star-full subtreeT of Γ not containing v, we write mv,T := mv,u, where u is any vertex of T .

6.1. The positive local tropicalization is contained in the splice fan.

In this subsection we show that the only points in ∆n−1 contained in the positive local tropicalizationTrop>0 S(Γ) are those included in ι(Γ). We exploit the terminology and convexity results stated in Section 5.

As expected, the Hamm determinant conditions imposed on the system S(Γ) from Definition 2.13 play acrucial role in determining Trop>0 S(Γ). The next lemma will be used extensively throughout this section:

Lemma 6.4. Fix a node v of Γ and let e, e′, e′′ be three distinct edges of StarΓ(v). Fix w ∈ (R≥0)n and

suppose that the admissible exponent vectors mv,e,mv,e′ ,mv,e′′ from (2.6) satisfy:

(6.1) w ·mv,e < w ·mv,e′ and w ·mv,e < w ·mv,e′′ .


Then, zmv,e = inw(f) for some f in the linear span of {fv,i : i = 1 . . . , δv − 2}. If, in addition, w satisfies

(6.2) w ·mv,e < w ·m for each m ∈δv−2⋃

i=1

Supp(gv,i),

then zmv,e = inw(F ) for some series F in the linear span of {Fv,i : i = 1 . . . , δv − 2}. In particular,w /∈ TropS(Γ).Proof. We let e1, . . . , eδv be the edges adjacent to v and assume that eδv−2 := e, eδv−1 := e′ and eδv := e′′.

Using the Hamm determinant conditions, we build a basis {f ′v,i}δv−2

i=1 for the linear span of {fv,i}δv−2i=1 where

f ′v,i := zmv,ei + ai z

mv,e′ + bi zmv,e′′ for each i ∈ {1, . . . , δv − 2}.

From (6.1) we conclude that inw(f′v,δv−2) = zmv,ei . Taking f := f ′

v,δv−2 proves the first part of the statement.

For the second part, the technique yields a new basis {F ′v,i}δv−2

i=1 for the linear span of {F ′v,i}δv−2

i=1 with

F ′v,i = f ′

v,i + g′v,i for each i ∈ {1, . . . , δv − 2},where each g′v,i is a linear combination of {gv,j}δv−2

j=1 . Condition (6.2) then ensures that

inw(F′v,δv−2) = inw(f

′v,δv−2) = zmv,e .

Thus, the series F = F ′v,δv−2 satisfies the required properties. In particular, the ideal inw(S(Γ)) contains

the monomial zmv,e and so w /∈ TropS(Γ) by definition. �

Next, we state the main theorem in this section, which yields one of the required inclusions in Theorem 6.2when choosing T = Γ. More precisely:

Theorem 6.5. For every star-full subtree T of Γ, we have ∆T ∩Trop>0 S(Γ) ⊆ ι(T ).

Proof. Recall from (5.10) that ∆T is the convex hull of the set of leaves ∂ T of T , viewed in ∆n−1 via themap ι. We proceed by induction on the number of nodes of T , which we denote by p. If p = 0, then T iseither a vertex or an edge of Γ, and ∆T = ι(T ). For the inductive step, assume p ≥ 1 and pick a node v ofT . Let T1, . . . , Tδv be the branches of T adjacent to v, as in Definition 5.3. We use the point ι(v) to performa stellar subdivision of ∆T , giving a decomposition ∆T =

⋃

λ∈∂ T τλ, where

(6.3) τλ := conv({ι(v)} ∪ ι(∂ T r {λ})) for all λ ∈ ∂ T.

By Lemma 5.8, τλ is a simplex of dimension |∂ T |− 1. Proposition 6.7 below shows that τλ ∩Trop>0 S(Γ)lies in the boundary of τλ. In turn, Proposition 6.8 ensures that

∂τλ ∩ Trop>0 S(Γ) ⊆⋃

1≤i≤δvi6=j

∆[Ti,v],

where Tj is the unique branch of T adjacent to v and containing the leaf λ. Thus, combining this with theinductive hypothesis applied to all star-full subtrees [Ti, v] of Γ with i ∈ {1, . . . , δv} gives

�(6.4) ∆T ∩Trop>0 S(Γ) ⊆δv⋃

i=1

(∆[Ti,v] ∩Trop>0 S(Γ)) ⊆δv⋃

i=1

ι([Ti, v]) ⊆ ι(T ).

As a natural consequence of this result, we deduce one of the two inclusions required to confirm Theorem 6.2:

Corollary 6.6. The positive local tropicalization of S(Γ) is contained in the support of the splice fan of Γ.

In the remainder of this subsection, we discuss the two key propositions used in the proof of Theorem 6.5.We start by showing that the relative interior of a top-dimensional simplex τλ from (6.3) obtained from thestellar subdivision of ∆T induced by a node v of T does not meet Trop>0 S(Γ). Lemma 6.4 plays a centralrole. The task is purely combinatorial and the difficulty lies in how to select the triple of admissible exponentvectors required by the lemma, that is compatible with the given input face of τλ.

Proposition 6.7. For every star-full tree T ⊂ Γ and every λ ∈ ∂ T , we have τλ◦ ∩ Trop>0 S(Γ) = ∅, where

τλ is the simplex from (6.3).


Proof. Let u be the unique node of T adjacent to λ, and denote by T ′1, . . . , T

′δu

the branches of T adjacentto u. We assume that T ′

δu= {λ} and pick any w ∈ τλ

◦. The definition of τλ allows us to write w uniquely as

w = αv ι(v) +

δu−1∑

j=1

wj where wj :=∑

µ∈∂ T ′

j

αµ,j ι(µ) for all j ∈ {1, . . . , δu − 1},

and αv, αµ,j > 0 for all µ ∈ ∂ T ′j .

We analyze the series Fu,i defining the system S(Γ) at u and use Lemma 6.4 to confirm that zmu,λ is thew-initial form of a series in the linear span of {inw(Fu,i) : i = 1, . . . , δu − 2}. Indeed, a direct computationtogether with (2.11) ensures that for each monomial zm appearing in gu,i we have:

(6.5) αv ι(v) ·mu,λ = αvℓu,v|wv|

< αv ι(v) ·m and wj ·mu,λ =∑

µ∈∂ T ′

jr∂ Γ

αµ,jℓu,µ|wµ|

≤ wj ·m for 1 ≤ j < δu.

Furthermore, the inequality in (6.5) involving wj is strict whenever ∂ T ′j * ∂ Γ. Thus, w ·mu,λ < w ·m for

each zm appearing in any series gu,i, as required by condition (6.2) of Lemma 6.4.To finish, we must find two branches T ′

j1and T ′

j2of T adjacent to u with j1, j2 < δv satisfying

(6.6) w ·mu,λ < w ·mu,T ′

j1and w ·mu,λ < w ·mu,T ′

j2.

The notation mu,T was introduced in Remark 6.3.Our choice will depend on the nature of u. If u = v we pick any j1, j2 < δv. On the contrary, if u 6= v we

take T ′j1

to be the unique branch of T adjacent to u containing v, and let T ′j2

be an arbitrary third branch.After relabeling, we may assume j1 = 1 and j2 = 2. In both cases, a direct computation yields

(6.7) αv ι(v) ·mu,λ = αv ι(v) ·mu,T ′

2≤ αv ι(v) ·mu,T ′

1.

Note that the inequality is strict when u 6= v. Furthermore,

(6.8) (w1+w2)·mu,λ < (w1+w2)·(mu,T ′

1+mu,T ′

2) and wj ·mu,λ = wj ·mu,T ′

1= wj ·mu,T ′

2for 3 ≤ j < δv.

Adding up (6.7) and (6.8) yields (6.6), as we wanted. This concludes our proof. �

Our next result is central to the inductive step in the proof of Theorem 6.5:

Proposition 6.8. Let T be a star-full subtree of Γ, and let v be a node of T . Denote by T1, . . . , Tδv thebranches of T adjacent to v. Let L be a proper subset of ∂ T which is not included in any ∂ Ti, and set

(6.9) P := conv({ι(v)} ∪ ι(L)).

Then, P is a simplex of dimension |L| and P◦ does not meet Trop>0 S(Γ).Proof. By Lemma 5.8 we have ι(v) ∈ (∆T )

◦. In addition, Proposition 5.10 (2) implies that P is a simplex

of the expected dimension. It remains to show that P◦ ∩ Trop>0 S(Γ) = ∅. For each j = 1, . . . , δv, we set

(6.10) Lj := L ∩ ∂ Tj and τj = R≥0〈conv(ι(Lj))〉.By definition, we have τj = {0} if Lj = ∅. Moreover, each τj is a simplicial cone of dimension |Lj |. Ourassumptions on L and a suitable relabeling of the branches Tj (if necessary) ensure the existence of someq ∈ {2, . . . , δv} with Lj 6= ∅ for all j ∈ {1, . . . , q} and Lj = ∅ for all j ∈ {q + 1, . . . , δv}.

We argue by contradiction and pick w ∈ P◦ ∩ Trop>0 S(Γ). Since P is a simplex, we write w as

(6.11) w = αv ι(v) +

q∑

j=1

wj with αv > 0 and wj ∈ τj r {0} for all j.

Lemma 6.9 below implies that Lj = ∂ Tj for all j ≤ q. Since L ( ∂ T , it follows that q < δv.Next, we use Lemma 6.4 to confirm that zmv,Tδv is the w-initial form of a series in the linear span of

{inw(Fv,i)}δv−2i=1 , which contradicts our assumption w ∈ Trop>0 S(Γ). First, a simple inspection shows that

ι(v) ·mv,Tδv= ι(v) ·mv,T2

= ι(v) ·mv,T1=

dv|wv|

, (w1 + w2) ·mv,δv < (w1 + w2) · (mv,T1+mu,T2

) and

wj ·mv,δv = wj ·mv,T1= wj ·mv,T2

for all j ∈ {3, . . . , q}.Combining these expressions yields w ·mv,Tδv

< w ·mv,T1and w ·mv,Tδv

< w ·mv,T2since αv > 0.


Figure 3. Building a maximal sub-branch of Tj avoiding Lj starting from a suitable leafλ ∈ ∂ TjrLj and moving inwards towards v, as in the proof of Lemma 6.9. Here, s ≤ δv′−2and r = δu′′ − 1.

To finish we compare the w-weight of zmv,Tδv with that of any exponent m in the support of a fixedgv,i. For each j ∈ {1, . . . , q} we write wj :=

∑

µ∈∂ Tjαµ,j ι(µ) with αµ,j > 0 for all µ. Then, the defining

properties of gv,i imply

(6.12) ι(v) ·mv,Tδv< ι(v) ·m and wj ·mv,Tδv

=∑

µ∈Ljr∂ Γ

αµ,jℓv,µ|wµ|

≤ wj ·m for 1 ≤ j ≤ q.

Note that the inequality for j ∈ {1 . . . , q} is strict whenever Lj * ∂ Γ.Combining both parts in (6.12) yields w ·mv,Tδv

< w ·m whenever zm appears in gv,i. This verifies thesecond hypothesis required for Lemma 6.4, contradicting our choice of w ∈ Trop>0 S(Γ). �

Lemma 6.9. For each j ∈ {1, . . . , δ}, let Lj be the set of leaves of Tj as in (6.10). If P◦∩Trop>0 S(Γ) 6= ∅,then Lj is either empty or equal to ∂ Tj.

Proof. We argue by contradiction and assume ∅ ( Lj ( ∂ Tj , so in particular |∂ Tj| > 1. We break theargument into four combinatorial claims, guided by Figure 3. The left-most picture informs the discussionfor Claims 1 and 2. The central picture refers to Claim 3, and the right-most picture illustrates Claim 4.Throughout, we fix w ∈ P◦ ∩ Trop>0 S(Γ) and write w as in (6.11) where wk =

∑

µ∈Lkαµ,k ι(µ) for each

k ∈ {1, . . . , q}, with αµ,k > 0 for all µ, k.First, we pick λ ∈ ∂ Tj rLj furthest away from v in the geodesic metric on T . Let v′ be the unique node

of Tj adjacent to λ. The condition |∂ Tj| > 1 ensures that v′ 6= v. This maximality restricts the nature ofthe node v′. More precisely,

Claim 1. The node v′ is an end-node of Tj.

Proof. We consider all branches T ′k of Tj adjacent to v′ and containing neither v nor λ. Our goal is to show

that |∂ T ′k| = 1 for all k. We argue by contradiction.

If |∂ T ′k| > 1, then by the maximality of the distance between v and λ we have ∂ T ′

k ⊆ Lj. We considerthe series defining S(Γ) at v′, and the admissible exponent vectors at v′ associated to λ, v and T ′

k. We claimthat the weight w satisfies

(6.13) w ·mv′,λ < w ·mv′,T ′

k, w ·mv′,λ < w ·mv′,v and w ·mv′,λ < w ·m

for each m in the support of some gv′,i. This cannot happen by Lemma 6.4 since w ∈ Trop>0 S(Γ).To prove the inequalities in (6.13) we analyze the contributions of each summand of w as in the proof

of Proposition 6.8. Our reasoning is similar to that of Proposition 6.7 replacing u by v′. First, we observe:

(6.14) αv ι(v) ·mv′,λ = αvℓv′,v

|wv|= αv ι(v) ·mv′,T ′

k< αv ι(v) ·mv′,v.

In turn, for each p ∈ {1, . . . , q}, the weight wp satisfies

(6.15) wp ·mv′,λ =∑

µ∈Lpr∂ Γ

αµ,pℓv′,µ

|wµ|≤ wp ·mv′,T ′

k.


Furthermore, the inequality is strict for p = j since Lj ( ∂ Tj . Combining (6.14) with (6.15) yields the leftinequality in (6.13). Similarly, we have

wp ·mv′,λ ≤ wp ·mv′,v for all p ∈ {1, . . . , q}.This inequality combined with (6.14) proves the central inequality in (6.13).

Finally, the properties defining the series gv′,i combined with (6.14) and (6.15) imply that for each mono-mial zm appearing in a fixed gv′,i we have

αv ι(v) ·mv′,λ < αv ι(v) ·m and wp ·mv′,λ ≤ wp ·m for all p = 1, . . . , q.

Thus, the right-most inequality in (6.13) is also valid. ⋄Claim 2. We have StarTj

(v′) ∩ Lj = ∅. In particular, none of the leaves of Tj adjacent to v′ can lie in Lj.

Proof. The claim follows by the same line of reasoning as Claim 1, working with the exponents mv′,v, mv′,λ

and mv′,T ′

kfor any other branch T ′

k of Tj adjacent to v′. If ∂ T ′k ⊆ Lj , the inequalities (6.13) will remain

valid, and this will contradict w ∈ Trop>0 S(Γ). ⋄As a consequence of Claim 2 we conclude that λ is part of a branch of Tj avoiding Lj with at least one

node. Let T ′ be a maximal branch of Tj with this property and furthest away from v. We claim that T ′ = Tj .To prove this, we argue by contradicting the maximality of T ′. We let u be the node of Tj adjacent to T ′

and u′ be the node of T ′ adjacent to u, as seen in the center of Figure 3. Next, we analyze the intersectionsbetween Lj and the leaves of all relevant branches of Tj adjacent to u. We treat two cases, depending onthe size of each such branch, starting with singleton branches:

Claim 3. None of the leaves of Tj adjacent to u belongs to Lj.

Proof. Pick a leaf µ of Tj adjacent to u. If µ ∈ Lj, a similar calculation to that of Claim 1 for the seriesFu,i replacing λ by T ′ and T ′

k by µ confirms that

w ·mu,T ′ < w ·mu,µ, w ·mu,T ′ < w ·mu,v, and w ·mu,T ′ < w ·mwhenever m ∈ Supp(gu,i). This cannot happen by Lemma 6.4 since w ∈ Trop>0 S(Γ). ⋄

Next, consider a non-singleton branch T ′′ of Tj adjacent to u and not containing v, with T ′′ 6= T ′. Asin the right-most picture in Figure 3, we let u′′ be the unique node of T ′′ adjacent to u in Γ and we letT ′′1 , . . . , T

′′r be the branches of T ′′ adjacent to u′′ that do not contain v. Then:

Claim 4. For all k, we have either ∂ T ′′k ⊆ Lj or ∂ T ′′

k ∩ Lj = ∅.Proof. If ∂ T ′′

k * Lj, picking λ ∈ ∂ T ′′k r Lj will produce a maximal branch of T ′′ not meeting Lj. This

branch will be further away from v than T ′ was, unless it equals T ′′k . Thus, we conclude ∂ T ′′

k ∩ Lj = ∅. ⋄The previous claim yields a stronger identity, namely:

(6.16) ∂ T ′′ ∩ Lj = ∅.To prove the latter, we first assume that ∂ T ′′ ⊆ Lj . We consider the series of S(Γ) determined by the nodeu. Replacing the roles of v′, λ and T ′

k in (6.13) by u, T ′ and T ′′, respectively, the same proof techniquefrom Claim 1 yields

w ·mu,T ′ < w ·mu,T ′′ w ·mu,T ′ < w ·mu,v and w ·mu,T ′ < w ·mfor each m in the support of any fixed gu,i. Lemma 6.4 then shows that w /∈ Trop>0 S(Γ), contradictingour original assumption on w. From here it follows that ∂ T ′′ * Lj , so by Claim 4 we can find some k with∂ T ′′

k ∩ Lj = ∅.Now, if ∂ T ′′ ∩ Lj 6= ∅, once again, Claim 4 yields ∂ T ′′

p ⊆ Lj for some p 6= k. Analyzing the equations ofS(Γ) at the node u′′, and replacing v′, λ and T ′

k in (6.13) by u′′, T ′′k and T ′′

p , respectively, we conclude:

w ·mu′′,T ′′

k< w ·mu′′,T ′′

pw ·mu′′,T ′′

k< w ·mu′′,v and w ·mu′′,T ′′

k< w ·m

for eachm in the support of a given gu′′,i. Lemma 6.4 then forces w /∈ Trop>0 S(Γ), leading to a contradiction.Therefore, (6.16) holds.

To finish, we observe that Claim 3 combined with (6.16) above contradicts the maximality of T ′, sincethe convex hull of all branches adjacent to u and not containing v will be a branch of Tj strictly containingT ′ and not meeting Lj. From here it follows that T ′ = Tj, which cannot happen since Lj 6= ∅. �


6.2. Boundary components of the extended tropicalization.

In this subsection, we characterize the boundary strata of the extended tropicalization of the germ (X, 0)defined by S(Γ) (see Remark 3.5). These strata are determined by the positive local tropicalization ofthe intersection of X with each coordinate subspace of Cn. This will serve two purposes. First, it willshow by combinatorial methods that (X, 0) is a two-dimensional complete intersection with no boundarycomponents. Second, it will help us prove the remaining inclusion from Theorem 6.5. The latter is thesubject of Subsection 6.3.

We start by setting up notation. Throughout, we write σ := (R≥0)n and fix a positive-dimensional proper

face τ of σ. Since N(∂Γ) ≃ Zn by choosing the basis {wλ : λ ∈ ∂ Γ} from Section 2, we define

(6.17) Lτ := {µ ∈ ∂ Γ : wµ is a ray of τ}.

We let k be the dimension of τ and consider the natural projection of vector spaces

(6.18) pτ : Rn → Rn/〈τ〉 ≃ Rn−k.

By abuse of notation, whenever wλ /∈ τ , we identify wλ with its image in Rn−k under pτ .

Definition 6.10. Given a series f ∈ C{{zλ : λ ∈ ∂ Γ}}, we let f τ be the series obtained from f by setting

all zλ with λ ∈ Lτ to be zero. We view f τ ∈ C{{zλ : λ ∈ τ⊥ ∩ Zn}} as a series in the n − k variables in{zλ : λ ∈ ∂ Γr Lτ}. We call it the τ-truncation of f .

Definition 6.11. We let Xτ be the intersection of the germ (X, 0) defined by S(Γ) with the dense torus in

the coordinate subspace of Cn associated to τ . This new germ is defined by the vanishing of the τ -truncationsof all series in S(Γ). The positive local tropicalization of S(Γ) with respect to τ is defined as the positive local

tropicalization of Xτ in Rn−k. Following [43, Section 12], we denote it by Trop>0(S(Γ), τ) .

Our first result generalizes Lemma 6.4, when some, but not all, admissible monomials at a fixed node ofΓ have trivial τ -truncations. It will simplify the computation of each Trop>0(S(Γ), τ).

Lemma 6.12. Fix a positive-dimensional proper face τ of σ, a node v of Γ and some w ∈ pτ (σ). Assume

that some τ-truncated series in {f τv,i}δv−2

i=1 is not identically zero and that one of these conditions hold:

(1) the system involves at most δv − 2 admissible monomials and there exists an edge e adjacent to vwith mv,e /∈ N〈zλ : λ ∈ Lτ 〉 satisfying w ·mv,e < w ·m for each m in the support of any (gv,i)

τ ; or(2) the system involves exactly δv − 1 monomials, and we have two distinct edges e, e′ of StarΓ(v) with

mv,e,mv,e′ /∈ N〈zλ : λ ∈ Lτ 〉 such that w ·mv,e < w ·mv,e′ and w ·mv,e < w ·m for each m in thesupport of any fixed (gv,i)

τ .

Then, zmv,e is the w-initial form of a series in the linear span of {(Fv,i)τ}δv−2

i=1 and w /∈ Trop>0(S(Γ), τ).

Proof. The proof follows the same reasoning as that of Lemma 6.4, considering the truncations of the newbasis {F ′

v,i}δv−2i=1 obtained by ordering the edges adjacent to v in a convenient way. If the conditions of (1)

hold, we pick any two edges e′, e′′ with mv,e′ ,mv,e′′ ∈ N〈zλ : λ ∈ Lτ 〉 and order the edges adjacent to v sothat eδv−2 := e, eδv−1 := e′ and eδv := e′′. In this situation, the statement follows since

inw((F′v,δv−2)

τ ) = inw((f′v,δv−2)

τ ) = zmv,e .

Similarly, if the conditions of (2) hold, we pick e′′ to be the unique edge with mv,e′′ ∈ N〈zλ : λ ∈ Lτ 〉. Thesame method produces the desired result for the series (F ′

v,δv−2)τ . �

Here is the main result of this section:

Theorem 6.13. Let τ be a positive-dimensional face of σ with λ ∈ Lτ . Let v be the unique node of Γadjacent to λ. Then:

(1) If dim τ = 1, then Trop>0(S(Γ), τ) ⊆ R>0〈pτ (wv)〉.(2) If dim τ ≥ 2, then Trop>0(S(Γ), τ) = ∅.


Proof. We fix k = dim τ . Since Trop>0(S(Γ), τ) is a fan in (R>0)n−k, we can determine the positive tropical-

ization by its intersection with the standard simplex ∆n−k−1 by following the proof strategy of Theorem 6.5.Since the vector pτ (wv) lies in (R>0)

n−k, we set

(6.19) w′v :=

pτ (wv)

|pτ (wv)|∈ (∆n−k−1)

◦

and perform a stellar subdivision of this standard simplex using w′v.

If dim τ = 1, Proposition 6.14 below ensures that for every simplex ρ of the stellar subdivision that meets(∆n−2)

◦ we have ρ◦ ∩ Trop>0(S(Γ), τ) = ∅ unless ρ is zero-dimensional. The only remaining simplex is{pτ (wv)}, so item (1) holds. In turn, when dim τ = 2, the same proposition applies also when ρ is a point.This forces Trop>0(S(Γ), τ) = ∅, as stated in item (2). �

Proposition 6.14. Let τ , λ and v be as in Theorem 6.13 and Lτ be as in (6.17). Fix a proper subset L of∂ Γr Lτ . Let w′

v be as in (6.19) and set ρ := conv ({w′v} ∪ {wµ : µ ∈ L}) ⊂ ∆n−k−1 where k = dim τ . If

|L ∪ Lτ | > 1, then

(6.20) ρ◦ ∩ Trop>0(S(Γ), τ) = ∅.Proof. We follow the proof strategy of Proposition 6.8 for T = Γ. We let T1, . . . , Tδv be the branches of Γadjacent to v, with T1 = {λ}. For each j ∈ {2, . . . , δv} we set

(6.21) Lτ,j := Lτ ∩ ∂ Tj , Lj := L ∩ ∂ Tj and ρj := R≥0〈pτ (wµ) : µ ∈ Lj〉 ⊂ Rn−k.

If Lj = ∅, we declare ρj = {0}. Since 1 < |L∪Lτ | < n, upon relabeling the branches T2, . . . , Tδv if necessary,we can find a unique q ∈ {2, . . . , δv} with Lj ∪ Lτ,j 6= ∅ for all j ∈ {1, . . . , q}, and Lj ∪ Lτ,j = ∅ for j > q.

We argue by contradiction and pick w ∈ ρ◦ ∩Trop>0(S(Γ), τ). Since ρ is a simplex and T1 ⊂ Lτ , we have

(6.22) w = αvw′v +

q∑

j=2

wj with αv > 0 and wj ∈ ρj◦ for all j ∈ {2, . . . , δv}.

In particular, we know that wj = 0 if and only if ρj = {0}.Lemma 6.15 below ensures that Lj ∪ Lτ,j = ∂ Tj for all j ≤ q. From here we conclude that 1 < q < δv,

so ∂ T1 ( L ∪ Lτ and ∂ Tδv ∩ (L ∪ Lτ ) = ∅. This implies that the τ -truncation of zmv,Tδv is non-zero, so

the τ -truncated series {(fv,i)τ}δv−2i=1 are not all identically zero. In turn, since zmv,T1 = z

dv,λ

λ , the system ofτ -truncations involves at most δv − 1 admissible monomials.

We claim that zmv,Tδv is the w-initial form of a series in the linear span of {(Fv,i)τ}δv−2

i=1 , which contradictsour assumption that w ∈ Trop>0(S(Γ), τ). We prove this claim by a case-by-case analysis. Each case matchesone of the two possible settings of Lemma 6.12.

First, we assume that the system of τ -truncations {(fv,i)τ}δv−2i=1 involves exactly δv − 1 monomials. In

particular, (zmv,T2 )τ 6= 0. This forces both ρ2 6= {0} and w2 ·mv,T2> 0 since L2 ∪ Lτ,2 = ∂ T2. We consider

the admissible exponent vectors mv,T2and mv,Tδv

, and fix any zm appearing in some gv,i with non-zeroτ -truncation. A direct calculation reveals the following inequalities and identities:

(6.23)αvw

′v ·mv,Tδv

= αvw′v ·mv,T2

= αvdv

|pτ (wv)|< αvw

′v ·m, w2 ·mv,Tδv

= 0 < w2 ·mv,T2,

0 ≤ w2 ·m and wj ·mv,Tδv= wj ·mv,T2

= 0 ≤ wj ·m for all j = 3 . . . , q.

Thus, the conditions for Lemma 6.12 (2) hold, as we wanted.

Finally, if the τ -truncated system {(fv,i)τ}δv−2i=1 has at most δv−2 admissible monomials, the result follows

again by Lemma 6.12 (1) since the inequalities involving mv,Tδvand m in (6.23) remain valid. �

Our next result is analogous to Lemma 6.9. We prove it using the same techniques.

Lemma 6.15. Fix Lτ,j, Lj and ρj as in (6.21). If ρ◦ ∩Trop>0(S(Γ), τ) 6= ∅, then the set Lj ∪Lτ,j is eitherempty or it equals ∂ Tj.

Proof. We use the notation established in the proof of Proposition 6.14 and pick w ∈ ρ◦ ∩Trop>0(S(Γ), τ).We write w as in (6.22). We argue by contradiction, assuming 0 < |Lj ∪ Lτ,j| < |∂ Tj |.

First, note that we can find a branch T ′ of Γ contained in Tj with ∂ T ′ ∩ (Lj ∪ Lτ,j) = ∅. For example,any leaf µ of ∂ Tj not in L ∪ Lτ will produce such a branch. Since Tj is finite, we can choose the branch T ′


Figure 4. Given a branch Tj of Γ adjacent to v and a fixed proper subset of leaves L of∂ Γr{λ} meeting ∂ Tj properly, we find a node u in Tj with distΓ(u, v) maximal so that thebranches T ′

1, . . . , T′r have a given property, but T ′

r+1, . . . , T′δu−1 do not. This construction is

the main ingredient in the proof of Lemma 6.15.

to be maximal with respect to the condition ∂ T ′ ∩ (Lj ∪Lτ,j) = ∅ and furthest away from v in the geodesicmetric on Γ. Let u be the unique node of Γ adjacent to T ′. Since (Lj ∪ Lτ,j) 6= ∅, we know that u 6= v.

Assume that there are r branches adjacent to u with this maximality property, and denote them byT ′1 = T ′, . . . , T ′

r as in Figure 4. We let T ′r+1, . . . , T

′δu−1 be the remaining branches of Γ adjacent to u and not

containing λ. The maximality of both T ′ and distΓ(u, v) combined with the condition u 6= v implies thatr < δu − 1 and ∂ T ′

i ⊆ Lj ∪ Lτ,j for each i ∈ {r + 1, . . . , δu − 1}. Indeed, if the latter were not the case, wecould find a branch T ′′ of Γ inside T ′

i with the same properties as T ′ but further away from v.

Note that since (zmu,T ′

1 )τ 6= 0, we know that the collection {(fu,i)τ}δu−2i=1 is not identically zero. We

claim that zmu,T ′

1 is the initial form of a series in the linear span of {(Fu,i)τ}δu−2

i=1 . This will contradict ouroriginal assumption that w ∈ Trop>0(S(Γ), τ). We prove our claim by analyzing three cases, depending onthe number of admissible monomials in each series that remain in its τ -truncation.

First, assume that fu,i = (fu,i)τ for each i ∈ {1, . . . , δu − 2}. Then, after replacing each gu,i with its

τ -truncation, we are in the setting of Lemma 6.4, where we view (R≥0)n−k ⊂ Rn

≥0 by adding zeros as

complementary coordinates. In this situation, the condition ∂ T ′1 ∩ L = ∅ and the defining properties of gu,i

ensure that

(6.24) w ·mu,T ′

1< w ·mu,v, w ·mu,T ′

1< w ·mu,T ′

δu−1and w ·mu,T ′

1< w ·m

for each zm appearing in some (gu,i)τ . The aforementioned lemma implies our claim regarding z

mu,T ′1 .

Second, assume the τ -truncated series {(fu,i)τ}δu−2i=1 involve at most δu− 2 admissible monomials. In this

case, the claim follows by Lemma 6.12 (1) since the inequalities involving mu,T ′

1and m in (6.24) remain valid

in this scenario.Finally, assume that the τ -truncated series at u involves exactly δu − 1 admissible monomials. In this

situation, up to relabeling of T ′r+1, . . . , T

′δu−1 we know that exactly one of the monomials with exponent

mu,v or mu,T ′

δu−1vanish under τ -truncation. The inequalities involving mu,T ′

1, m and the surviving among

mu,v or mu,T ′

δu−1in (6.24) ensure that the hypotheses required by Lemma 6.12 (2) hold. �

Theorem 6.13 has the following two important consequences:

Corollary 6.16. The intersection of the germ defined by the system S(Γ) with any coordinate subspace Hof Cn of codimension at least two is just the origin.

Proof. Let H ′ be the minimal coordinate subspace of Cn contained in H with X ∩H ′ 6= {0}, and considerthe face τ of (R≥0)

n associated to H ′. It follows that Xτ 6= ∅, but this cannot happen since dimXτ =dimTrop>0(S(Γ), τ) by Proposition 3.11 (1) and Trop>0(S(Γ), τ) = ∅ by Theorem 6.13 (2). �

Corollary 6.17. The germ (X, 0) defined by the system S(Γ) intersects each coordinate hyperplane along agerm of a curve. All its irreducible components meet the dense torus of the corresponding hyperplane.

Proof. We fix a coordinate hyperplane {zλ = 0} and let τ be the cone generated by wλ in Rn. Counting thenumber of equations defining Y := X ∩ {zλ = 0} in Cn, we see that dimY ≥ 1 by Krull’s principal idealtheorem. By construction, Xτ = Y ∩ (C∗)n−1. Since any face τ ′ of (R≥0)

n properly containing τ satisfies


Xτ ′ = ∅ by Corollary 6.16, it follows from Proposition 3.11 (1) that dimXτ = dimTrop>0(S(Γ), τ) and,furthermore, no component of Y lies in the toric boundary of Cn−1. However, dimTrop>0(S(Γ), τ) ≤ 1by Theorem 6.13 (1), so dimXτ = 1. �

In turn, the last corollary has two consequences. First, it confirms that the germ defined by S(Γ) is acomplete intersection, and second, it shows that equality must hold in Theorem 6.13 (1).

Corollary 6.18. The germ (X, 0) defined by the system S(Γ) is a two-dimensional complete intersection.Each of its irreducible components meets the dense torus (C∗)n non-trivially in dimension two.

Note that this result allows a priori for the germ (X, 0) to have several irreducible components. We willsee in Corollary 7.10 that (X, 0) is in fact irreducible.

Corollary 6.19. Let λ be a leaf of Γ and let v be the unique node of Γ adjacent to it. If τ = R≥0〈wλ〉, thenTrop>0(S(Γ), τ) = R>0〈pτ (wv)〉.Proof. The result follows by combining Theorem 6.13 (1), Corollary 6.17, and the equality between thedimensions of Xτ and Trop>0(S(Γ), τ) stated in Proposition 3.11 (1). �

Finally, Corollary 6.18 and Proposition 3.8 combined characterize the local tropicalization of (X, 0) → Cn:

Corollary 6.20. The local tropicalization of the germ (X, 0) is the Euclidean closure of Trop>0 S(Γ) in Rn.

6.3. The splice fan is contained in the local tropicalization of the splice type system.

In this subsection, we prove the remaining inclusion in Theorem 6.2. Our arguments are purely combi-natorial, and rely on the balancing condition for pure-dimensional local tropicalizations (see Remark 3.17).Corollary 6.18 confirms that such condition holds for the positive tropicalization of the germ defined byS(Γ). Furthermore, the proofs in this section imply that no proper two-dimensional subset of Trop>0 S(Γ)is balanced.

We start by stating the main result in this section. Its proof will be broken into several lemmas andpropositions for clarity of exposition.

Theorem 6.21. For every splice diagram Γ, we have ι(Γ) ⊆ TropS(Γ) in (R≥0)n.

Proof. By Lemma 6.22 below we know that ∆n−1 ∩TropS(Γ) is a 1-dimensional polyhedral complex andι(v) ∈ Trop>0 S(Γ) for some node v of Γ. We claim that, in fact, ι(u) ∈ Trop>0 S(Γ) for each node u of Γ.

We prove this claim by induction on the distance between u and v (recall that Γ is connected). If u = vthere is nothing to show. For the inductive step, pick a node u with distΓ(u, v) = k ≥ 1 and assume that theclaim holds for each node u′ of Γ with distΓ(u

′, v) = k − 1. Let u′ be the unique node of [u, v] adjacent tou. Then, ι(u′) ∈ Trop>0 S(Γ) by our inductive hypothesis. Proposition 6.24 below and Corollary 6.20 yield:

ι(u) ∈ ι(StarΓ(u′)) ∩ (R>0)

n ⊂ TropS(Γ).The desired inclusion ι(Γ) ⊂ Trop>0 S(Γ) follows by combining Proposition 6.24 with the identity

ι(Γ) =⋃

v node of Γ

StarΓ(v). �

Our first lemma ensures that the image of some node of Γ lies in the positive tropicalization of S(Γ).Lemma 6.22. The intersection ∆n−1 ∩TropS(Γ) is a 1-dimensional polyhedral complex. Furthermore,there exists a node v of Γ with ι(v) ∈ ∆n−1 ∩Trop>0 S(Γ).Proof. By Corollary 6.18 we know that TropS(Γ) is a fan of pure dimension two, and so ∆n−1 ∩TropS(Γ)is a pure 1-dimensional polyhedral complex. To conclude, we must find a node v with ι(v) ∈ Trop>0 S(Γ).

Since Trop>0 S(Γ) ⊂ ι(Γ) by Corollary 6.6, we have two possibilities for any w ∈ ∆n−1 ∩Trop>0 S(Γ):either w = ι(v) for some node v of Γ or w ∈ ι([u, u′]

◦) for two adjacent vertices u, u′ of Γ. In the second

situation, Lemma 6.23 ensures that ι([u, u′]) ⊂ ∆n−1 ∩TropS(Γ). Since one of u or u′ must be a node of Γand each node maps to (∆n−1)

◦ under ι, the claim follows. �

Our next lemma is central to the proof of both Theorem 6.21 and Lemma 6.22. It describes the possibleintersections between the local tropicalization of S(Γ) and the edges of Γ embedded in ∆n−1 via the map ι.The balancing condition for positive local tropicalizations plays a prominent role.


Lemma 6.23. Let u, u′ be two adjacent vertices of Γ. If ι([u, u′]◦) intersects Trop>0 S(Γ) non-trivially, then

ι([u, u′]) ⊆ TropS(Γ).

Proof. It suffices to show that ι([u, u′]◦) ⊆ TropS(Γ). We argue by contradiction and fix a point w ∈

ι([u, u′])rTrop>0 S(Γ). We fix a standard homeomorphism ϕ : [0, 1]→ [u, u′] with ϕ(0) = u and ϕ(1) = u′,and write w = ϕ(t0) for a unique t0 ∈ [0, 1]. Since TropS(Γ) is closed in (R≥0)

n, we can find a pair a, b ∈ [0, 1]satisfying a < t0 < b and such that the open segment (ι(ϕ(a)), ι(ϕ(b))) avoids Trop>0 S(Γ). Furthermore,we pick (a, b) to be the maximal open interval in [0, 1] containing t0 with this property. A contradiction willarise naturally if we prove that a = 0 and b = 1.

By symmetry, it suffices to show that a = 0. We argue by contradiction, and assume a > 0, soι(ϕ(a)) ∈ (∆n−1)

◦ by construction. The maximality of (a, b) combined with Corollary 6.20 ensures thatι(ϕ(a)) ∈ Trop>0 S(Γ). Recall from Corollary 6.6 that ∆n−1 ∩Trop>0 S(Γ) ⊆ ι(Γ). Since ι is an injectionby Theorem 5.11, the balancing condition for Trop>0 S(Γ) forces the star of ι(ϕ(a)) in Trop>0 S(Γ) to be a2-dimensional cone. This cannot happen since ι(ϕ(a, b)) avoids Trop>0 S(Γ). �

Our next result plays a prominent role in the induction arguments used to prove Theorem 6.21. Onceagain, the balancing condition for local tropicalizations becomes crucial.

Proposition 6.24. Let v be a node of Γ with ι(v) ∈ Trop>0 S(Γ). Then, ι(StarΓ(v)) ⊆ TropS(Γ).

Proof. After refinement if necessary, we may assume that τ := R≥0 ι(v) is a ray of Trop>0 S(Γ). Given avertex u of Γ adjacent to v, we consider the 1-dimensional saturated lattice

Λ :=Zn ∩ R〈wu, wv〉Zn ∩ R〈wv〉

.

Let a vector wu|τ ∈ Zn ∩ R≥0 ι([u, v]) be such that its natural projection to Λ generates this lattice. Inparticular, we can write wu|τ uniquely as

(6.25) wu|τ := αu ι(u) + βu ι(v)

for some αu, βu ∈ Q and with αu > 0.We let u1, . . . , uδv be the vertices of Γ adjacent to v. The balancing condition for Trop>0 S(Γ) at τ

combined with Theorem 6.5 ensures the existence of non-negative integers {k1, . . . , kδv} (i.e., the tropicalmultiplicities) satisfying:

(6.26)

δv∑

i=1

ki wui |τ ∈ Zn ∩ R〈wv〉.

Moreover, by Definition 3.16, (ι([uj , v]))◦intersects Trop>0 S(Γ) non-trivially in a neighborhood of ι(v) if

and only if kj 6= 0.Since Trop>0 S(Γ) is pure of dimension two and ι(v) ∈ Trop>0 S(Γ), we know that kj0 > 0 for some

j0 ∈ {1, . . . , δv}. We claim that, furthermore, all k1, . . . , kδv are positive integers. The inclusion ι(StarΓ(v)) ⊂TropS(Γ) follows by combining this statement with Lemma 6.23.

To prove our claim, we consider the following linear equation in k1, . . . , kδv that is equivalent to (6.26):

(6.27)

δv∑

j=1

(kj αuj) ι(uj) = β ι(v) with β ∈ Q.

First, we argue that this system admits a unique solution (k1, . . . , kδv ) ∈ Qδv for all β. Uniqueness followsdirectly since αuj

> 0 for all j and the set {ι(u1), . . . , ι(uδv )} is linearly independent by Proposition 5.10 (1).Second, we claim that for β 6= 0, any solution to (6.27) has kj 6= 0 for all j. By homogeneity, we may

assume β = 1. Then, Proposition 5.7 (2) and the linear independence of {ι(uj)}δvj=1 force the coefficients

kjαujin (6.27) to be the ones used to write ι(v) as an element of (∆StarΓ(v))

◦. Since αuj

> 0 for all j wehave kj > 0 for all j, as we wanted.

To finish we argue that (6.26) has a solution with kj0 > 0 by the balancing condition. This forces β 6= 0in (6.27), and so kj 6= 0 for all j by the previous discussion. This concludes our proof. �


7. Splice type singularities are Newton non-degenerate

In this section we discuss the Newton non-degeneracy of splice type systems (see Theorem 7.1), followingthe original framework introduced by Khovanskii (see Definition 4.2). This property only involves the initialforms of the generators of the system S(Γ), as opposed to the initial ideals of the ideal 〈S(Γ)〉 generated bythe elements of the system S(Γ). In turn, Theorem A.1 ensures the w-initial forms of (Fv,i)v,i generate thew-initial ideal of 〈S(Γ)〉, for each w in the positive local tropicalization of S(Γ), thus giving an alternativeproof of Theorem 6.21. Furthermore, Newton non-degeneracy implies that condition (3) of Proposition 3.11holds for every cell in ι(Γ), thus showing that the splice fan of Γ is a standard tropicalizing fan for the germdefined by the system S(Γ), in the sense of Definition 3.13. This is the content of Corollary 7.7. In turn,Corollary 7.11 gives an alternative proof of a recent theorem of de Felipe, Gonzalez Perez and Mourtada [8],which resolves any germ of reduced plane curve by one toric morphism after a reembedding in a higherdimensional complex affine space. The section concludes with an open question regarding embedding dimen-sion of splice type surface singularities preserving the Newton non-degeneracy property (see Question 7.13).

We start by setting up notation. For each weight vector w ∈ (R>0)n we define:

(7.1) Jw := 〈inw(Fv,i) : v node of Γ, i = 1, . . . , δv − 2〉 ⊂ C[zλ1, . . . , zλn

].

and let Zw be the subscheme of Cn defined by Jw . Newton non-degeneracy will be certified by showing

that Zw ∩ (C∗)n is smooth whenever w ∈ Trop>0 S(Γ). This is the content of the main result of this section:

Theorem 7.1. The splice type system S(Γ) is a Newton non-degenerate complete intersection system.

Proof. This follows from Lemmas 7.2, 7.3 and 7.4 below. Indeed, monomial curves are smooth outside theorigin by Theorem 2.22 and for each node u of Γ, the components of the scheme Zwu

are Pham-Brieskorn-Hamm singularities and they can only meet at a coordinate subspace of Cn. Thus, Zwu

∩(C∗)n is smooth. �

The next three lemmas characterize the ideal Jw associated to points in the relative interior of each cellin the polyhedral complex ι(Γ).

Lemma 7.2. Let u be a node of Γ and let w = wu. Let Γ0 be the star of Γ at u, viewed as a splice diagramwith inherited weights around u. Then:

(1) Jw = 〈inw(fv,i) : v node of Γ, i = 1, . . . , δv − 2〉;(2) Zw is a complete intersection of dimension two;(3) Zw has no component in the boundary of (C∗)n, so Jw is monomial-free;(4) all components of Zw are images of C under a (torus-translated) monomial map whose domain is a

Pham-Brieskorn-Hamm complete intersection in Cδu determined by Γ0;(5) the intersection of any two distinct components of Zw lies outside of (C∗)n.

Proof. Item (1) follows from the identity inwu(Fv,i) = inwu

(fv,i) for all v and i that is valid due to (2.11). Toprove the remaining items, we let k be the number of nodes of Γ adjacent to u and T1, . . . , Tk the branchesof Γ adjacent to u, each Tj containing the corresponding node uj adjacent to u. Set Γi = [Ti, u] and expressJw as a sum of the following k + 1 ideals:

J0 = 〈inw(fu,j) : j = 1, . . . , δu − 2〉, Ji = 〈inw(fv,j) : v node of Γi, j = 1, . . . , δv − 2〉, for i = 1, . . . , k.

Note that the generators of each Ji with i 6= 0 lie in C[zλ : λ ∈ ∂ Ti]. In particular, no variable zλ with λadjacent to u appears in them. To fix notation, we write these leaves as λ1, . . . , λl with k + l = δu.

For simplicity, we write Z = Zw . We start by characterizing the components of Z. Each ideal Ji withi ∈ {1, . . . , k} defines an end-curve Ci determined by the splice diagram Γi rooted at u. In turn, Theorem 2.22ensures that each of the (gi-many) components of Ci admits a (torus-translated) monomial parameterizationof the form

(7.2) C[zλ : λ ∈ ∂uΓi]→ C[ti] where zλ 7→ c(u)λ,i t

ℓu,λ/gii ,

with c(u)λ,i 6= 0 for all λ, i. Thus, each component of C1× . . .×Ck can be parameterized using Ck by combining

these (torus-translated) monomial maps.


Substituting the expressions from (7.2) for the chosen component of each Ci into the generators of J0shows that the closure of each component of Z can be parameterized using a component of a splice typesingularity defined by the diagram Γ0. Note that the Hamm determinant conditions at u arising from S(Γ0)agree with those determined by the generators of J0 up to multiplying the columns corresponding to thebranches Γi, i = 1, . . . , k by non-zero constants. Therefore, each component of Z is the image of a (torus-translated) monomial map from Cδu restricted to a Pham-Brieskorn-Hamm complete intersection definedby Γ0. In particular, no component of Z lies in a coordinate subspace of Cn, so Jw is monomial-free. Thisproves both (3) and (4).

Since Z is equidimensional of dimension two and it is defined by n − 2 polynomial equations, it is acomplete intersection. This proves (2). It remains to address (5) when Z is not irreducible.

To determine the intersection of two distinct components (say, Z1 and Z2) of Z we exploit the parameter-ization described earlier. Let I ⊂ {1, . . . , k} be the collection of indices i for which the projection of Z1 andZ2 to Spec(C[ti]) do not agree. If I = ∅, then the two components are parameterized using the same Pham-Brieskorn-Hamm system of equations associated to Γ0 and the same (torus-translated) monomial map. Thiscannot happen since such germs are irreducible and Z1 6= Z2.

Next, assume |I| ≥ 1. In this setting, the projection of Z1∩Z2 to C∂uΓi for each i ∈ I lies in the intersectionof two components of the end-curve Ci, which can only be origin of C∂uΓi by Theorem 2.22. This means thatZ1 ∩ Z2 lies in the coordinate subspace of Cn defined by the vanishing of all zλ with λ ∈ ⋃

i∈I ∂uΓi. Thisconcludes our proof. �

Lemma 7.3. Let u be a node of Γ adjacent to a leaf λ and pick w ∈ [ι(u), ι(λ)]◦. Then, Zw is a cylinder

over a monomial curve in Cn−1 with gcd(ℓλ,µ : µ ∈ ∂ Γr {λ}) many components and Jw is monomial-free.

Proof. We prove that Jw defines the cylinder over the end-curve Cλ associated to the rooted splice diagramΓλ. A simple inspection shows that for all nodes v of Γ and each i ∈ {1, . . . , δv−2}, the initial form inw(Fv,i)agrees with the corresponding polynomial hv,i defining Cλ (see Definition 2.21). Indeed, if we write w as

(7.3) w = α ι(λ) + (1− α) ι(u) with 0 < α < 1,

then the defining properties of gv,i ensure that inw(Fv,i) = inw(fv,i) for all v, i. In turn, combining this factwith Proposition 2.11 and Lemma 2.18 yields

w ·mv,e ≥ (1− α)ℓu,v|wu|

for each edge e in StarΓ(v).

Furthermore, equality holds if and only if e * [v, λ]. Thus, inw(fv,i) is obtained from fv,i by dropping theadmissible monomial at v pointing towards λ. In particular, these initial forms do not involve zλ.

The above discussion shows that J ′w := Jw ∩ C[zµ : µ ∈ ∂ Γ r {λ}] defines Cλ. The general point of this

curve lies in (C∗)n−1 by Theorem 2.22, so J ′w is monomial-free. In turn, the same result confirms that J ′

w

defines a reduced complete intersection, smooth outside the origin, and with gcd(ℓλ,µ : µ ∈ ∂ Γr {λ}) manyirreducible components. Since Jw is obtained from J ′

w by base change to C[zλ : λ ∈ ∂ Γ], and both idealsadmit a common generating set, the result follows. �

Given two adjacent nodes u and u′ of Γ, we let T ′ be the branch of Γ adjacent to u′ containing u. Similarly,we let T be the branch of Γ adjacent to u and containing u′. Our final lemma is analogous to Lemma 7.3:

Lemma 7.4. Let u and u′ be two adjacent nodes of Γ and pick w ∈ [ι(u), ι(u′)]◦. Then, Zw is isomorphic

to a product of two monomial curves in C|∂ T ′| ×C|∂ T |. Furthermore, the number of irreducible componentsof Zw equals gcd(ℓ′u′,λ : λ ∈ ∂ T ′) gcd(ℓ′u,µ : µ ∈ ∂ T ) and Jw is monomial-free.

Proof. We write w = αwu +(1−α)wu′ with 0 < α < 1 and follow the same proof-strategy as in Lemma 7.3.The conditions on α guarantee that for each node v in T ′, inw(Fv,i) is obtained from fv,i by droppingthe admissible monomial at v pointing towards u′. This observation and the symmetry between u andu′ determine a partition of the generating set of Jw , where each initial form inw(Fv,i) for any node v ofT ′ (respectively, in T ) only involves the leaves of T ′ (respectively, of T ). Thus, each set determines theend-curves for the diagrams [T ′, u′] and [T, u], rooted at u′ and u, respectively.

By construction, Jw defines the product of these two monomial curves. Since each of them is a completeintersection in their respective ambient spaces, the same is true for Jw . The number of components isdetermined by Theorem 2.22. Since Zw meets (C∗)n, we conclude that Jw is monomial-free. �


The next result is a direct consequence of Theorem A.1 and the previous three lemmas. It gives analternative proof of Theorem 6.21 since for any w in ι(Γ)

◦, these lemmas confirm that Jw is monomial-free.

Corollary 7.5. For any strictly positive vector w in the splice fan of Γ, we have inw(S(Γ)) = Jw .

Proof. By construction, we have Jw ⊆ inw(S(Γ)) for each w ∈ (∆n−1)◦ ∩ ι(Γ). In addition, Lemmas 7.2, 7.3

and 7.4 imply that the generators of Jw form a regular sequence in the ring of convergent power seriesC{zλ : λ ∈ ∂ Γ} near the origin. Thus, Theorem A.1 implies that Jw = inw(S(Γ)) as ideals of C{zλ : λ ∈ ∂ Γ}for each w ∈ Qn ∩R≥0〈ι(Γ)〉. Finally, since these ideals are constant when we consider all rational points inthe relative interior of any fixed edge of ι(Γ), the same must be true for all points in these open segments. �

Remark 7.6. The geometric information collected in Lemmas 7.3 and 7.4 combined with Corollary 7.5 de-termine the tropical multiplicities of Trop>0 S(Γ). Using Remark 2.6 and Definition 3.16, we have:

(1) if τ = R≥0 ι([λ, u]) for a node u of Γ, then τ has multiplicity 1du,λ

gcd(ℓu,µ : µ ∈ ∂ Γr {λ});(2) if τ = R≥0 ι([u, v]) for two adjacent nodes u and v of Γ, then τ has multiplicity

1

du,vdv,ugcd(ℓu,λ : λ ∈ ∂ Γ, u ∈ [λ, v]) gcd(ℓv,λ : λ ∈ ∂ Γ, v ∈ [λ, u]).

This information completes the characterization of Trop>0 S(Γ) as a tropical object, i.e., as a weightedbalanced polyhedral fan. We use this data in Section 8 when discussing how to recover Γ from its splice fan.

Our next statement follows naturally from Proposition 3.11 and Corollary 7.5:

Corollary 7.7. The splice fan of Γ is a standard tropicalizing fan for the germ (X, 0) defined by S(Γ).We conclude this section by showing how we can use their local tropicalizations to recover some known

facts about splice type systems and their associated end-curves from [32, 33]. Our first statement discussesthe intersection of the initial degenerations of the germ defined by S(Γ) will suitable codimension-2 coordinatesubspaces of Cn:

Lemma 7.8. Let w be a vector which has strictly positive coordinates and lies on the embedded splice diagramι(Γ), and let λ and µ be two leaves of Γ in different branches of ι(Γ) adjacent to w. Then, the system

{

inw(Fv,i) = 0, v is a node of Γ, i = 1, . . . , δv,

zλ = zµ = 0

has 0 as its only solution.

Proof. If w = ι(u) is the (normalized) weight vector corresponding to a node u of Γ, this statement can beproved using the arguments outlined in the proof of [32, Theorem 2.6], in particular, on page 710. However,a closer look reveals that this reasoning can also be used for any point w in ι(Γ) ∩ (∆n)

◦. �

Our second statement gives a stronger version of Corollary 2.24:

Corollary 7.9. Assume that the conditions of Lemma 7.8 hold and consider the map Fw : Cn → Cn−2

obtained from the collection {inw(Fv,i)}v,i, ordered appropriately. Then, the restriction of this map to thecodimension-2 subspace L = {zλ = zµ = 0} of Cn is surjective.

Proof. By Lemma 7.8, we see that the fiber of the restriction of Fw to L over the origin of Cn−2 is 0-dimensional. By upper semicontinuity of fiber dimensions, the generic fiber of Fw|L is also 0-dimensional.Since dimL = n− 2, the map Fw|L is dominant.

Since Fw|L is defined by weighted homogeneous functions, it admits a projectivization as a map betweenweighted projective spaces. But a dominant projective map must be surjective. Thus, as an affine map,Fw|L is surjective as well. �

Next, we recover Theorem 2.15 (originally due to Neumann and Wahl) by combining Corollary 6.16 withthe following result:

Corollary 7.10. The singularity defined by the splice type system S(Γ) is isolated. In particular, it is alsoirreducible.


Proof. Let (X, 0) → Cn be the germ defined by S(Γ). By Corollary 6.16, we know that X∩(C∗)n is dense inX . Since S(Γ) is a Newton non-degenerate complete intersection system, X ⊂ Cn admits an embedded toricresolution (see, e.g., Khovanskii [18, Section 2.7] and Oka [35, Chapter III, Theorem (3.4)]). In particular,for a suitable subdivision Σ of the splice fan of X , the corresponding toric morphism π : XΣ → Cn inducesan embedded resolution of the pair (X,Cn). But as we saw in Theorem 6.2, the local tropicalization of S(Γ)intersects the boundary of the non-negative orthant only along the canonical basis elements. It follows fromthis that the morphism π is an isomorphism outside the origin, i.e., π is a resolution of (X, 0). We concludethat the singularity at the origin is isolated. As (X, 0) is moreover a complete intersection of dimension twoby Corollary 6.18, it is automatically irreducible. �

We end this section by showing how to use our results to resolve singularities of complex plane curvesby combining a re-embedding of C2 with toric modifications. The fact that such a resolution was possiblewas proven by Goldin and Teissier [14] in the irreducible case and recently by de Felipe, Gonzalez Perez andMourtada [8] in full generality.

Corollary 7.11. Let (X, 0) → C2 be a germ of a reduced complex analytic plane curve. Then, the ambientgerm (C2, 0) can be holomorphically re-embedded into a suitable higher-dimensional germ (Cn, 0) in such away that the induced germ (X, 0) → Cn can be resolved by a single toric modification of Cn if this last spaceis endowed with its standard toric structure.

Proof. The result is a direct consequence of Theorem 7.1. Indeed, we consider a completion (X, 0) → C2 of

the input germ (X, 0) → C2 in the sense of [12, Definition 1.4.15]. By construction, (X, 0) is also a germof reduced plane curve, it contains (X, 0) as a subgerm and it admits an embedded resolution (that is, a

modification π : S → C2 such that S is smooth and the total transform of X on S has normal crossings)

such that the strict transform of X intersects all the leaf components of the exceptional divisor (i.e., thosecorresponding to leaves of the dual graph). Moreover, the modification π : S → C2 can be chosen so

that each leaf component is intersected by exactly one irreducible component of the strict transform of X.Since all the components of (X, 0) are principal divisors on (C2, 0), and (C2, 0) has an integral homologysphere link, Neumann and Wahl’s end-curve theorem [33, Theorem 4.1 (3)] guarantees the existence of aholomorphic embedding φ : (C2, 0)→ (Cn, 0) such that (φ(C2), 0) is a splice type singularity and such that

the components of (φ(X), 0) are the intersections of (φ(C2), 0) with all coordinate hyperplanes of (Cn, 0).Applying Theorem 7.1 to this smooth splice type singularity proves the desired statement for (X, 0) → C2.

Indeed, any regular subdivision of the tropicalizing fan of (φ(C2), 0) in Rn≥0 from Corollary 7.7 may be

extended to a subdivision of the non-negative weight orthant Rn≥0 (viewed in Cn), which in turn provides a

toric modification of Cn which resolves (X, 0), and thus, also (X, 0). �

Remark 7.12. Results from [11, Section 5] allow us to describe the splice diagram associated to a plane curvesingularity in terms of the Newton-Puiseux series of its branches. Applying our findings to this splice diagramyields a concrete description of the local tropicalization of the embedding φ from the proof of Corollary 7.11in terms of standard combinatorial invariants of the given plane curve singularity.

The construction of splice type systems by Neumann and Wahl implies that the embedding dimension ofa splice type singularity is bounded above by the number of leaves on the associated splice diagram. If anedge ending in a leaf has weight one, then it can be removed without changing the associated class of splicetype singularities. But even if no such edge exists, the bound need not be attained for concrete examples.In particular, [33, Example 3] exhibits a hypersurface singularity Z(f) in C3 realizing a splice type surfacesingularity (X, 0) whose associate splice diagram Γ has six leaves.

A simple calculation reveals that the standard local tropicalization of this hypersurface is the cone over astar-shaped tree with three leaves (corresponding to the standard coordinate basis vectors) and a single nodew. The w-initial form of f is non-reduced so the Newton non-degeneracy condition fails for this hypersurfacepresentation, despite the fact that the splice type system S(Γ) defining (X, 0) is a Newton non-degeneratecomplete intersection system by Theorem 7.1. The following question arises naturally:

Question 7.13. Fix a splice diagram with n leaves satisfying the edge determinant and semigroup conditions.Assume that no edge of Γ ending in a leaf has weight one. Let (X, 0) be a splice type singularity associated toΓ. Can we embed (X, 0) in some (Cm, 0) with m < n while preserving the Newton non-degeneracy condition?


Figure 5. A splice diagram Γ which cannot be recovered uniquely from its splice fan, asin Remark 8.4. Here, d1 may take the values 1, 2, 4 and all tropical multiplicities of thesplice fan equal 4.

8. Recovering splice diagrams from splice fans

The construction of splice fans from splice diagrams raises a natural question: how much data about Γ canbe recovered from its splice fan, decorated with the tropical multiplicities? The main result of this sectiongives a positive answer to this question under the following coprimality restrictions, which are central to [33]:

Definition 8.1. We say a splice diagram Γ is coprime if the weights around each node of Γ are pairwisecoprime.

Our next result highlights the restrictions on the tropically weighted splice fan imposed by a coprimesplice diagram. Its proof will be postponed until the end of this section. Precise formulas for the tropicalmultiplicities are given in Remark 7.6.

Theorem 8.2. Let Γ be a coprime splice diagram. Then:

(1) for each node v of Γ the vector wv ∈ N(∂Γ) ≃ Zn is primitive;(2) all tropical multiplicities in Trop>0 S(Γ) equal one.

The following is the main result in this section. It ensures that coprime splice diagrams can be recoveredfrom their tropically weighted splice fans:

Theorem 8.3. Assume that all tropical multiplicities on the splice fan equal one. Then, there is a uniquecoprime splice diagram Γ yielding the given splice fan.

The rest of the section is devoted to the proof of these two results. A series of lemmas and propositionswill simplify the exposition.

Remark 8.4. Notice that the analog of Theorem 8.3 may fail if we drop the tropical multiplicity one restric-tions. This can be seen by looking at the example in Figure 5. For each choice of edge weight d1 = 1, 2 or4, the diagram Γ satisfies the semigroup and edge determinant conditions. Furthermore, all tropical multi-plicities on the 2-dimensional cones of the splice fan equal four. For each value of d1 we can choose systemsS(Γ) defining a germ in C4, whose local tropicalization is supported on the input splice fan.

Our first technical result will allow us to employ a pruning argument to prove Theorem 8.3. To this end,we use superscripts to indicate the underlying splice diagram considered for the computation of each linkingnumber. The absence of a superscript refers to Γ. The same notation will be used for weight vectors.

Proposition 8.5. Let [u, v] be an internal edge of a splice diagram Γ. Let T be the branch of Γ adjacent tou and containing v. Consider Γ′ = [u, T ] with weights around its nodes inherited from Γ. Then, the weightedtree Γ′ is a splice diagram, i.e., it satisfies the semigroup and the edge determinant conditions.

Proof. We only need to check that the semigroup conditions hold for Γ′. The linking numbers for Γ involvinga vertex v′ of Γ′ with v′ 6= u and a leaf λ of Γ can be obtained from those in Γ′ via:

(8.1) ℓv′,λ =

{

ℓΓ′

v′,λ if λ ∈ ∂ Γ′ ∩ ∂ Γ,

ℓΓ′

v′,uℓu,λ

du,votherwise .

Since the semigroup condition at each v′ holds for Γ, expression (8.1) implies the same is true for Γ′. �

Assume that u and v are adjacent nodes of Γ and let Γ′ be the associated splice diagram introducedin Proposition 8.5. Up to relabeling, we write ∂ Γr ∂ Γ′ = {λ1, . . . , λs} for some s. Consider the following


n× (n− s+ 1) matrix with integer entries in block form obtained from (8.1):

(8.2) A :=

ℓu,λ1/du,v... 0

ℓu,λs/du,v0 Idn−s

.

A direct computation yields the following identity for the tree Γ′:

Lemma 8.6. For each vertex v′ of Γ′ with v′ 6= u we have AwΓ′

v′ = wv′ .

Remark 8.7. Notice that AwΓ′

u =(∑

λ∈∂uΓℓu,λ/du,v

)Bar(∂uΓ; [u, v]) 6= wu.

Lemma 8.8. Fix a coprime splice diagram Γ. Let u, v be two adjacent nodes of Γ, and let Γ′ be the diagramfrom Proposition 8.5. Then, we have dv,u = gcd(ℓv,λ : λ ∈ ∂ Γ′ ∩ ∂ Γ).

Proof. The result follows by an easy induction on the number of nodes of Γ′. If Γ′ has two nodes, the resultholds by the coprimality of the weights around v. For the inductive step, we assume that v is adjacent to qleaves and k nodes other than u, denoted by {µ1, . . . , µq} and {v1, . . . , vk}, respectively. Then,

(8.3) gcd(ℓv,µ1, . . . , ℓv,µq

) = dv,u

k∏

j=1

dv,vj .

For each j ∈ {1, . . . , k} we let Tj be the branch of Γ′ adjacent to v and containing vj . Let Γ′j = [v, Tj ]

be the corresponding splice diagram with inherited weights. The inductive hypothesis on each Γ′j yields

dvj ,v = gcd(ℓvj ,µ : µ ∈ ∂ Γ′j r {v}). The identity ℓv,µ = (ℓvj ,µ/dvj ,v) (dv/dv,vj ) where µ ∈ ∂ Γ′

j r {v} gives

(8.4) gcd(ℓv,µ : µ ∈ ∂ Γ′j r {v}) =

dvdv,vj

.

The result follows by combining (8.3) and (8.4) with the coprimality of the weights at v. �

Proof of Theorem 8.2. We prove the statement by induction on the number of nodes of Γ, which we denoteby p. If p = 1, we let u be the unique node of Γ. Then, the coprimality condition

gcd(du,λi, dv,λj

) = 1 for i 6= j

implies that wu is a primitive vector. Furthermore, the formula in Remark 7.6 confirms that the tropicalmultiplicity associated to the edge [u, λ] equals one since

1

du,λgcd(du,λ

∏

γ 6=λ,µ

du,γ : µ ∈ ∂ Γr {λ}) = 1.

Next, assume p > 1 and let u be an end-node of Γ. Let {λ1, . . . , λs} be the leaves adjacent to u and letv be the unique node of Γ adjacent to u. We let Γ′ be the splice sub-diagram of Γ obtained from u and v,as in Proposition 8.5. The coprimality condition for Γ′ and our inductive hypothesis confirm that for eachvertex v′ of Γ′, wΓ′

v′ is a primitive vector in N(∂Γ′), and all tropical multiplicities of Trop>0 S(Γ′) are one.Since the gcd of all maximal minors of the matrix A equals 1 by the coprimality condition around u,

it follows that A maps primitive vectors in N(∂Γ′) to primitive vectors in N(∂Γ). This fact togetherwith Lemma 8.6 ensures that the vector wv′ is primitive whenever v′ 6= u is a node of Γ′. If v′ = u we have

wu =

s∑

i=1

dudu,λi

wλi+

dudu,v

∑

µ∈∂vΓ

ℓv,µdv,u

wµ,

where ∂vΓ is the set ∂u,[u,v]Γ from Definition 2.12. Since gcd(ℓv,µ : µ ∈ ∂vΓ} = dv,u by Lemma 8.8, andgcd(du/du,λi

: i = 1, . . . , s} = du,v, the pairwise coprimality of weights around u ensures that wu is aprimitive vector in N(∂Γ).


To finish, we compute the tropical multiplicities. Remark 7.6, Lemma 8.8 and the coprimality of weightsaround u implies that the multiplicity corresponding to the edge [u, λi] of Γ is one. Indeed, we have

gcd(ℓu,µdu,λi

: µ ∈ ∂ Γr {λi}) = gcd(gcd(du

du,λidu,λj

j = 1, . . . , s, j 6= i), gcd(ℓv,µdv,u

dudu,λi

du,v: µ ∈ ∂vΓ))

= gcd(gcd(du

du,λidu,λj

j = 1, . . . , s, j 6= i),du

du,λidu,v

) = 1.

If we pick an edge [v′, λj ] with j > s we get multiplicity one by the inductive hypothesis applied to Γ′

combined with (8.1) and the coprimality of the weights around u. More precisely,

gcd(gcd(ℓv′,λk

dv′,λk

: k = s+ 1, . . . , n, k 6= j), gcd(ℓv′,λi

dv′,λk

: i = 1, . . . , s)) =

gcd(gcd(ℓΓ

′

v′,λk

dv′,λk

: k = s+ 1, . . . , n, k 6= j),ℓΓ

′

v′,u

dv′,λk

gcd(ℓu,λi

du,v: i = 1, . . . , s)

︸︷︷︸

=1

) = 1.

Finally, the cone associated to an edge between two adjacent nodes u′, v′ of Γ will have tropical multiplicityone by Lemma 8.8 since gcd(ℓu′,λ/du′,v′ : λ ∈ ∂v′,[u′,v′]Γ) = gcd(ℓv′,µ/dv′,u′ : µ ∈ ∂u′,[u′,v′]Γ) = 1. �

Proof of Theorem 8.3. The combinatorial type of Γ is completely determined by intersecting ∆n−1 and thesplice fan. In turn, Theorem 8.2 (1) allows us to characterize the vector wu as the primitive vector associatedto the corresponding ray of the fan R≥0 ι(Γ). All that remains is to determine the weights around each nodeof Γ from this data. We do so by induction on the number of nodes of Γ, which we denote by p.

If p = 1, then the coprimality of the weights around the single node u of Γ determines each du,λiuniquely

as follows. By construction, the entries of wu are coprime and we have

du,λ = gcd((wu)µ : µ ∈ ∂ Γr {λ}).Next, assume p > 1 and fix an end-node u of Γ. Let λ1, . . . , λs be the leaves of Γ adjacent to u, and v be

the unique node of Γ adjacent to u. Let Γ′ be the tree obtained by pruning Γ from u, as in Proposition 8.5.The weights around u can be recovered uniquely from the splice fan of Γ. Indeed, write

(8.5) wu =

s∑

i=1

dudu,λi

wλi+

dudu,v

n∑

j=s+1

ℓΓ′

v,λj

dv,uwλj

.

Notice that gcd(ℓΓ′

v,λjdu/(du,vdv,u) : j = s + 1, . . . , n} = du/dv,u by Lemma 8.8. In turn, the coprimality

condition gives du,v = gcd(du/du,λi: i = 1, . . . , s) and du = lcm(du/du,λi

: i = 1, . . . , s). From this werecover all remaining s weights at u since du,λi

= du/(du/du,λi) for i ∈ {1, . . . , s}.

Next, for each node v′ of Γ with v′ 6= u, we use the full-rank matrix A from (8.2) to recover wΓ′

v′ ∈ Zn−s+1

uniquely from wv′ . Since wΓ′

u is a prescribed canonical basis element of N(∂Γ′) ≃ Zn−s+1, the set of vectors

{wΓ′

v′ : v′ is a node of Γ′} allows us to determine the splice fan of Γ′. The inductive hypothesis then uniquelyrecovers the splice diagram Γ′, and hence Γ has been fully determined. �

9. Intrinsic nature of splice type singularities

Theorem 6.2 shows that the local tropicalization of the germ defined by a given splice type system S(Γ) isindependent of the choice of admissible monomials and higher order terms used to define it. In fact, Neumannand Wahl proved that the set of splice type singularities defined by splice type systems with fixed admissiblemonomials associated to a given splice diagram is independent of these choices, both in the coprime settingand in the general case under a suitable equivariant hypothesis on the series gv,i collecting the higher orderterms of each series Fv,i. In this section we give a variant of their proof in the coprime case and we show byan example that without the equivariance hypothesis, the result no longer holds.

Here is the precise statement for coprime diagrams, which can be deduced from the equivariant case [32,Theorem 10.1]. For completeness, we include a direct proof:


Theorem 9.1. Let Γ be a coprime splice diagram with n leaves, and let

M := {zmv,e : v ∈ V ◦(Γ), e ∈ StarΓ(v)}be a complete set of admissible monomials for Γ. Let S(Γ)M be the set of all splice type systems {Fv,i =fv,i+gv,i : v, i} that can be constructed using the setM. Then, the set XM of all germs (X, 0) → Cn definedby the vanishing of a system in S(Γ)M is independent ofM.

Proof. LetM′ be a second complete set of admissible monomials. We proceed by induction on the size mof the set M rM′. If m = 0, there is nothing to show. For the inductive step, it suffices to analyze thecase when m = 1, i.e. when M and M′ differ by exactly one admissible monomial. Fix a pair (v0, e0) forwhich the corresponding monomials inM andM′ differ, and let mv0,e0 and m′

v0,e0 , be the exponent vectorsof those monomials. To prove the statement, we use the wv0 -filtration I• : I0 ⊇ I1 ⊇ I2 ⊇ . . . of ideals of thelocal ring O := C{z1, . . . , zn}, where(9.1) Id := {f ∈ O : wv0(f) ≥ d}.

Let {Fv,i : v, i} ∈ S(Γ)M. It determines a germ (X, 0) ∈ XM together with an embedding ϕ : X → Cn.We write Fv,i = fv,i + gv,i for each node v of Γ and i ∈ {0, . . . , δv − 2} as in (2.9) and (2.10). The wv0 -filtration I• restricted to X yields a filtration J• under the corresponding surjective map of local ringsϕ♯ : O ։ O(X,0), i.e.,

(9.2) Jk := ϕ♯(Ik) for all k ≥ 0.

By Lemma 9.2 below, there exist a ∈ C∗, g ∈ Idv+1 and h in the ideal of O spanned by {Fv,i : v, i},satisfying the equality

(9.3) zmv0,e0 − a zm′

v0,e0 = h+ g.

Furthermore, up to moving to g all higher-order contributions of terms in h coming from each Fv0,i, we mayassume that h =

∑

v,i av,iFv,i with av0,i ∈ C for each i ∈ {1, . . . , δv}.We prove the inclusion XM ⊆ XM′ by constructing an explicit system {F ′

v,i : v, i} in S(Γ)M′ whosevanishing set equals X . The reverse inclusion then follows by exploiting the symmetry between M andM′. We consider the set {F ′

v,i : v, i} with F ′v,i := Fv,i for each v 6= v0 and i ∈ {1, . . . , δv − 2}, whereas

F ′v0,i

:= f ′v0,i

+ g′v0,i for i ∈ {1, . . . , δv0 − 2} with

(9.4) f ′v0,i := (fv0,i − cv0,e0,iz

mv0,e0 ) + (cv0,e0,i/a)zm′

v0,e0 and g′v0,i := gv0,i − (cv0,e0,i/a) g.

and the series g ∈ Idv0+1 is the one in (9.3).

We claim that {F ′v,i} ∈ S(Γ)M′ . Indeed, by construction, each series g′v0,i lies in Idv0

, as required by (2.11).

In addition, the matrix of coefficients for the polynomials {F ′v0,i}i is obtained from the matrix (cv0,e,i)i,e

after rescaling the column labeled by e0. Thus, the Hamm determinant conditions of Definition 2.13 aresatisfied.

Combining (9.3) and (9.4) yields

F ′v0,i = Fv0,i +

cv0,e0,ia

h ∈ 〈Fv,j : v node of Γ, j = 1, . . . , δv − 2〉.

We use the expression of h given above to replace {F ′v,i : v, i} by a set generating the same ideal, i.e.,

(9.5) {Fv,i : v 6= v0, i = 1, . . . , δv − 2} ∪ {∑

1≤j≤δv0−2j 6=i

av0,j Fv0,j + (1 + av0,i)Fv0,i : i = 1, . . . , δv0 − 2}.

Since both {Fv,i : v, i} ∈ S(Γ)M and {F ′v,i : v, i} ∈ S(Γ)M′ determine complete intersection systems of

equations by Theorem 2.15, the (δv0 − 2)× (δv0 − 2)-matrix of scalars Id+(av0,i)i associated to the secondset in (9.5) must be invertible. From here it follows that the vanishing sets of both collections {F ′

v,i : v, i}and {Fv,i : v, i} agree. Thus, the germ X lies in XM′ , as we wanted to show. This concludes our proof. �

The following technical lemma gives a more precise version of the first half of the statement of [32, Theorem10.1]. Its proof follows the same reasoning, so we omit it here:


Figure 6. An example confirming the dependency of splice-type systems on the choice ofadmissible monomials.

Lemma 9.2. Fix two collections of admissible monomials M,M′ with |M rM′| = 1. Assume X ∈ XM,

and let zmv0,e0 ∈ M rM′ and zm′

v0,e0 ∈ M′ rM. Then, there exists a ∈ C∗ such that the restriction

(zmv0,e0 − azm′

v0,e0

)

|X∈ Jdv+1, where Jdv+1 is the ideal from (9.2).

As we mentioned earlier, if Γ is not coprime, analogous results to Theorem 9.1 and Lemma 9.2 can beproved under the condition that the higher order terms of each system (i.e., the terms in each gv,i) satisfyan equivariant condition under the action of a suitable finite abelian group, namely, the discriminant groupD(Γ) of a given plumbing graph with associated splice diagram Γ. For a precise statement, we refer to [33,Theorem 10.1]).

It is natural to ask whether this equivariant condition can be weakened. The next example shows thatthis is not the case.

Example 9.3. Consider the (non-coprime) splice diagram Γ from Figure 6 and pick two sets of admissiblemonomials for Γ that differ only in the choice of exponent vectors for the pair (v, [v, u]):

M := {z21 , z22 , z3z4, z23 , z24} ∪ {z31}, M′ := {z21 , z22 , z3z4, z23 , z24} ∪ {z1z2}.We claim that XM 6= XM′ , i.e., the elements of S(Γ)M and S(Γ)M determine different sets of subgerms of(Cn, 0). More precisely, we show that the germ in XM defined by the system

(9.6)

{

fu,1 := z21 + z22 + z3 z4,

fv,1 := z31 + z23 + z24 ,

in S(Γ)M cannot be a member of XM′ . To do so, it suffices to show that no power series associated to thenode v of Γ, that is, no power series of the form

F ′v,1 := b1 z

21z2 + b2 z

23 + b3 z

24 + gv,1,

with b1, b2, b3 ∈ C∗ and gv,1 satisfying (2.11) can be an element of the ideal (fu,1, fv,1) of the power seriesring C{z1, . . . , z4}.

We argue by contradiction and pick elements A1, A2 ∈ C{z1, . . . , z4} with(9.7) b1 z

21z2 + b2 z

23 + b3 z

24 + gv,1 = A1 fu,1 +A2 fv,1.

By construction, the wv-initial form on the left-hand side is b1 z21z2 + b2 z

23 + b3 z

24 and its wv-weight is 24.

We claim that the wv-initial form on the right-hand side of (9.7) may be written as

α1(z)(z21 + z22) + α2(z)(z

31 + z23 + z24),

for two wv-homogeneous polynomials α1(z), α2(z).We prove this claim by explicit computation, comparing the wv-weights of both summands and noticing

that inwv(fu,1) = z21 + z22 , inwv

(fv,1) = fv,1. Three situations can occur. First, if wv(A1fu,1) < wv(A2fv,1),then the wv-initial form on the right-hand side of (9.7) comes from the first summand, i.e., α1(z) = inwv

(A1)and α2(z) = 0. Similarly, if wv(A1fu,1) > wv(A2fv,1), then the second summand determines the wv-initialform on the right-hand side of (9.7), so α1(z) = 0 and α2(z) = inwv

(A2). Finally, if wv(A1fu,1) = wv(A2fv,1),the condition that the total wv-weight of A1fu,1 + A2fv,1 agrees with the wv-weight of F

′v,1 confirms that

both terms contribute to the wv-initial form, i.e. α1(z) = inwv(A1) and α2(z) = inwv

(A2).Comparing the wv-initial forms on both sides of (9.7) yields an identity of wv-homogeneous polynomials:

b1 z21z2 + b2 z

23 + b3 z

24 = α1(z)(z

21 + z22) + α2(z)(z

31 + z23 + z24).

Since the wv-weight on both sides equals 24, we conclude that α2(z) must be a constant. Evaluating bothsides at z1 = z2 = 0 forces b2 = b3 = α2(z), so in particular α2 := α2(z) ∈ C∗. We conclude from this that

b1 z21z2 = α1(z)(z

21 + z22) + α2 z

31 .


This identity implies that the function z31/(z21z2) = b1α

−12 is constant on V (z21 + z22) ∩ (C∗)2, which is false.

This contradiction confirms that XM 6= XM′ , as we wanted to show. ⋄

Acknowledgments

Maria Angelica Cueto was supported by an NSF postdoctoral fellowship DMS-1103857 and NSF StandardGrants DMS-1700194 and DMS-1954163 (USA). Patrick Popescu-Pampu was supported by French grantsANR-12-JS01-0002-01 SUSI, ANR-17-CE40-0023-02 LISA and Labex CEMPI (ANR-11-LABX-0007-01).Labex CEMPI also financed a one month research stay of the first author in Lille during Summer 2018.

Part of this project was carried out during two Research in triples programs, one at the Centre Interna-tional de Rencontres Mathematiques (Marseille, France, Award number: 1173/2014) and one at the CenterInternational Bernoulli (Lausanne, Switzerland). The authors would like to thank both institutes for theirhospitality, and for providing excellent working conditions.

The authors are very grateful to Jonathan Wahl for contributing the proof presented in the appendix ofthis paper.

Appendix A. Initial ideals and local regular sequences(by Jonathan Wahl)

In [32], the authors invoke a folklore lemma in commutative algebra in order to prove several of their maintheorems. This result involves regular sequences in a polynomial ring and their initial forms with respect tointeger weight vectors. As originally stated, [32, Lemma 3.3] is not quite-correct: the global setting mustbe replaced by a local one. This appendix provides a complete proof of this result in the local setting ofconvergent power series near the origin, a result we could not locate in the literature. This local version

agrees with the general framework of [32]. Throughout, we let n be a positive integer and let (O,m) denote

the local ring of convergent power series C{z1, . . . , zn} near the origin.

We start by stating our main result, namely, a reformulation of [32, Lemma 3.3] in the local setting. Itsproof will be given at the end of this appendix, after discussing a series of preliminary technical results. Notethat the same statement and proof will hold if O denotes the localization of the polynomial ring C[z1, . . . , zn]at the maximal ideal of the origin of Cn.

Theorem A.1. Let (f1, . . . , fs) be a finite sequence of elements in the maximal ideal m of O, and let J bethe ideal generated by them. Fix a positive weight vector w ∈ (Z>0)

n. Assume that (inw(f1), . . . , inw(fs)) isa regular sequence in O. Then:

(1) the sequence (f1, . . . , fs) is also regular, and(2) the w-initial ideal inw(J)O is generated by {inw(f1), . . . , inw(fs)}.

Remark A.2. As mentioned earlier, Theorem A.1 does not hold in the polynomial setting. For instance,(z1(1 − z1), z2(1 − z1)) is a regular sequence in the local ring C{z1, z2} but not in the polynomial ringC[z1, z2]. However, the sequence (z1, z2) of initial forms with respect to any weight vector w ∈ (Z>0)

2 isregular in both rings.

Remark A.3. The regularity of the sequence of w-initial forms is needed in Theorem A.1. As an example,fix n = 4, w = (1, 1, 1, 1), and consider the sequence (f1, f2) with

f1 := z21 + z42 − z33 and f2 := z1 z2 − z34 .

By construction, (f1, f2) is a regular sequence in O defining an isolated complete intersection surface singu-larity. The sequence of initial forms (inw(f1), inw(f2)) = (z21 , z1 z2) is not regular, and the w-initial ideal of(f1, f2)O is generated by inw(f1), inw(f2) and inw(z2 f1 − z1 f2) = −z2 z33 + z1 z

34 .

Throughout, we fix w ∈ (Z>0)n and an arbitrary sequence (f1, . . . , fs) of elements of the maximal ideal m.

We let J be the ideal generated by the fi’s. Consider the first few steps in the Koszul complex of O-modulesdetermined by it (see, e.g., [45, Sect. IV.A]):

(A.1) F :=⊕

1≤i<j≤s

O · eijd2 // E :=

⊕

1≤i≤s

O · eid1 // O // O/J.


The map d1 : E → O sends ei to fi for each i = 1, . . . , s and the kernel R of d1 is the module of relationsbetween the given generators of J . The morphism d2 : F → E sends eij to fj ei − fi ej , and its image is thesubmodule of “trivial relations” between (f1, . . . , fs). By definition, the image of d2 lies in R, so we view d2also as as a map d2 : F → R.

By a standard result in commutative algebra (see, e.g., [45, Prop.3, Chapter IV.A.2]) we have:

Proposition A.4. The sequence (f1, · · · , fs) of elements in m is regular in O if and only if the Koszulcomplex (A.1) is exact at E.

Since the definition of E does not depend on the order of the sequence (f1, . . . , fs), the following conse-quence arises naturally:

Corollary A.5. If (f1, . . . , fs) is a regular sequence in O, any reordering of it is also a regular sequence.

The weight vector w inducing the w-weight valuation (3.1) on O endows this ring with a weight filtration

by ideals (Ip)p≥0, where Ip := {g ∈ O : w(g) ≥ p}. Similarly, we can filter O via the ideals (mp)p≥0. Both

filtrations are cofinal since

(A.2) Idp ⊆ mp ⊆ Ip for all p ≥ 0,

where d is the maximum among all coordinates of w. It follows from this that the completions of Owith respect to both filtrations are canonically isomorphic. The completion induced by the m-adic filtration(mp)p≥0 is the ring of formal power series in n variables.

In a similar fashion, we can filter the modules E and F appearing in (A.1) via (Ep)p≥0 and (Fp)p≥0,respectively, by assigning the weights w(fi) and w(fi) + w(fj) to ei and eij , respectively. More precisely,

(A.3) Ep := {s∑

i=1

aiei : w(ai) ≥ p−w(fi) ∀i} and Fp := {∑

i<j

bijeij : w(bij) ≥ p−w(fi)−w(fj) ∀i, j}.

These choices ensure that the maps d1 and d2 from the Koszul complex (A.1) preserve the filtration. Inaddition, the module R of relations is filtered as well, via

(A.4) Rp := Ep ∩R.

We use these filtrations to define the w-initial forms on E and F . We state the definition for E, since theone for F is analogous. The definition for R is given by restriction.

Definition A.6. Given any g ∈ E with g 6= 0, we let p be the unique integer such that g ∈ Ep r Ep+1.An element g :=

∑si=1 ri ei ∈ Ep r Ep+1 satisfies w(ri) + w(fi) ≥ p for all i ∈ {1, . . . , s} and equality must

hold for some index i. Let I be the set of indices where equality is achieved. The w-initial form of g is

inw(g) :=∑

i∈I inw(ri)ei. We set inw(0) = 0.

By Proposition A.4, the regularity of the sequence (f1, . . . , fs) is equivalent to the surjectivity of the mapd2 : F → R induced by (A.1). We prove the latter in Lemma A.11, assuming the regularity of the sequenceof w-initial forms of all fi’s.

Our first two lemmas use the regularity assumptions for the sequence of w-initial forms to prove thesurjectivity of d2 : F → R by working with the filtrations of F and R described above.

Lemma A.7. Assume that the sequence (inw(f1), · · · , inw(fs)) is regular in O. Then, the morphism ofC-vector spaces ϕp : Fp/Fp+1 → Rp/Rp+1 induced by the morphism of O-modules d2 : F → R is surjectivefor all integers p ≥ 0.

Proof. We must show that modulo Rp+1, every element g of Rp is the image of an element of Fp/Fp+1 underthe map ϕp. If g = 0, there is nothing to show, so we assume g 6= 0. In particular, g lifts to an element inRp rRp+1, which we denote by g as well. We write g =

∑sj=1 rj ej .

Assume that inw(g) has k many terms, with k ∈ {1, . . . , s} (see Definition A.6). By Corollary A.5, wecan reorder the original sequence while preserving its regularity, and write inw(g) as

inw(g) =

k∑

j=1

inw(rj)ej with w(rj) + w(fj) = p for all j ∈ {1, . . . , k}.


We claim that g is congruent, modulo the image of ϕp, to an element of Rp whose w-initial form lies in theideal generated by {inw(f1), . . . , inw(fk−1)}. The original statement will follow by induction on k ≤ s.

Since∑s

j=1 rjfj = 0 by definition of R and w(rj)+w(fj) > p for j ∈ {k+1, . . . , s}, we conclude that theexpected w-initial form of

∑sj=1 rjfj must vanish, i.e.,

k∑

j=1

inw(rj) inw(fj) = 0.

Therefore, inw(rk) inw(fk) is zero modulo the ideal I := 〈inw(f1), . . . , inw(fk−1)〉O. Since the sequence(inw(f1), . . . , inw(fs)) is regular, we conclude that inw(rk) must lie in I.

Taking the w-weight value of rk and each fj into account we write inw(rk) as

inw(rk) =k−1∑

j=1

aj inw(fj),

where aj is either 0 or a non-zero w-weighted homogeneous polynomial with w(aj) = p−w(fk)−w(fj) ≥ 0for all j ∈ {1, . . . , k − 1}. It follows from this that the element

r′k := rk −k−1∑

j=1

ajfj

satisfies w(r′k) > p− w(fk), so r′k ek ∈ Ep+1. Simple arithmetic manipulations give a new formula for g, i.e,

(A.5) g =

s∑

j=1

rj ej =

k−1∑

j=1

(rj + aj fk) ej +

k−1∑

j=1

aj(fj ek − fk ej)

︸︷︷︸

:=h

+r′k ek +

s∑

j=k+1

rj ej .

By construction, it follows that h = ϕp(∑k−1

j=1 ajejk) ∈ ϕp(Fp/Fp+1). Furthermore, inw(g) only involves

terms in the first of the four summands on the right-hand side of (A.5) since the last two summands lie inEp+1. This establishes the claim. �

Lemma A.8. Let J be the ideal of O generated by {f1, . . . , fs} and assume that (inw(f1), · · · , inw(fs)) isa regular sequence in O. If g ∈ J has w-weight equal to p ∈ N, then g admits an expression of the formg =

∑si=1 aifi, where w(aifi) ≥ p for all i. In particular, inw(g) belongs to the ideal of O generated by

{inw(f1), . . . , inw(fs)}.

Proof. Since g ∈ J , we may write g as g =∑s

i=1 bi fi with bi ∈ O for each i. Consider

p′ = min{w(bifi) : i = 1, . . . , s}.Assume that this weight is achieved at k many terms, which we can fix to be {b1 f1, . . . , bk fk} upon reordering.If p′ ≥ p we have w(aifi) ≥ p for all i and equality must hold for some i by definition of p. From here it

follows that p′ = p, so inw(g) =∑k

j=1 inw(aj) inw(fj), as we wanted to show.

On the contrary assume that p′ < p. We claim that we can find an alternative expression g =∑s

j=1 b′j fj

where the corresponding minimum weight p′′ := min{w(b′j fj)} satisfies p′′ ≥ p′ and the number of summandsrealizing p′′ is strictly smaller than k. An easy induction combined with the fact that p′, p ∈ Z≥0 will thenyield a new expression for g with p′ ≥ p, as in our previous case.

It remains to prove the claim. Since p′ < p, the terms in g with w-weight p′ must cancel out, i.e.,∑k

j=1 inw(bj) inw(fj) = 0. As in the proof of Lemma A.7, the fact that (inw(f1), . . . , inw(fs)) is a regularsequence in O ensures that

inw(bk) =

k−1∑

j=1

cj inw(fj),

where cj is either zero or a w-homogeneous polynomial with w(cj) = p′−w(fj)−w(fk) ≥ 0. It follows from

here that the element b′k := bk −∑k−1

j=1 cj fj has weight w(b′k) > w(bk), so w(b′k fk) > p′.


An arithmetic manipulation allows us to rewrite g as follows:

(A.6) g =

k−1∑

j=1

(bj + cj fk) fj + b′k fk +

s∑

j=k+1

bj fj

By construction, the terms with minimum w-weight only appear in the first of the three summands onthe right-hand side of (A.6). Furthermore, the corresponding minimum weight p′′ satisfies p′′ ≥ p′ sincew(bjfj) ≥ p′ for all j and w(cj fkfj) ≥ p′ for j < k. This confirms the validity of our claim. �

A standard commutative algebra result (see, e.g., [3, Lemma 10.23]) combined with Lemma A.8 yields:

Lemma A.9. Assume that the sequence (inw(f1), · · · , inw(fs)) is regular in O. Then, the map d2 : F → R offiltered modules induces a surjection between their completions relative to the filtrations (Fp)p≥0 and (Rp)p≥0

respectively. More precisely, lim←−F/Fp ։ lim←−R/Rp.

We let F and R be the m-adic completions of F and R respectively, which can be computed withstandard methods. Indeed, by [3, Theorem 10.13], we have

(A.7) F ≃ F ⊗O O and R ≃ R⊗O O.The double inclusions in (A.2) allow us to compare the completions in Lemma A.9 induced by (Fp)p≥0 and

(Rp)p≥0, with F and R, respectively. More precisely,

Lemma A.10. Assume that the sequence (inw(f1), · · · , inw(fs)) is regular in O. Then, the completionsappearing in Lemma A.9 agree with the m-adic ones, i.e.

(A.8) lim←−F/Fp ≃ lim←−F/mpF ≃ F ⊗O O and lim←−R/Rp ≃ lim←−R/mpR ≃ R⊗O O.Proof. We let ℓ := max{w(fj) : j = 1, . . . , s}. It suffices to prove the first isomorphism on each side of (A.8),since the remaining ones appear in (A.7). By (A.3), we have

Ep :=⊕

i

Ip−w(fi) ei and Fp :=⊕

i<j

Ip−w(fi)−w(fj) eij for each p ≥ 0.

It follows from here that IpE ⊆ Ep ⊆ Ip−ℓE and IpF ⊆ Fp ⊆ Ip−2ℓF for each p ≥ 0. Combining theseinclusions with (A.2) yields:

(A.9) Edp+ℓ ⊆ IdpE ⊆ mpE ⊆ Ep and Fdp+2ℓ ⊆ IdpF ⊆ m

pF ⊆ Fp for each p ≥ 0.

The inclusions appearing on the right of (A.9) ensure that the filtrations (mpF )p≥0 and (Fp)p≥0 are cofinalin F . Thus, they yield isomorphic completions. This proves the first isomorphism in (A.8).

Next, consider the filtration Rp from (A.4). First, notice that mpR ⊆ Rp by (A.9). To finish, we claim

the existence of some k ≥ 0 for which Rdp+(dk+ℓ) ⊆ mpR for all p ≫ 0. Indeed, by the Artin-Rees Lemma

(see, e.g., [3, Theorem 10.10]), there exists an integer k ≥ 0 satisfying

R ∩mpE = m

p−k(R ∩mkE) for all p ≥ k.

Therefore, combining this fact with property (A.9) we obtained the desired inclusion:

Rdp+(dk+ℓ) = R ∩ Ed(p+k)+ℓ ⊂ R ∩mp+kE = m

p(R ∩mkE) ⊂ m

pR.

We conclude that (mpR)p≥0 and (Rp)p≥0 are cofinal filtrations in R, so they yield isomorphic completions. �

We let M be the cokernel of the map d2 : F → R given by (A.1), and we let M be its m-adic completion.Lemma A.10 yields the following result:

Lemma A.11. Assume that the sequence (inw(f1), · · · , inw(fs)) is regular in O. Then, M = 0 and M = 0.In particular, the Koszul complex (A.1) is exact at E.

Proof. By standard commutative algebra (see, e.g.,[45, Corollaire 2, Chap. II.A.5]) we know that O is a

flat O-module. Therefore, taking m-adic completion is an exact functor. Since F → R is surjective (by

combining Lemmas A.9 and A.10) it follows that M = 0.

By [3, Theorem 10.17], the kernel of the canonical morphism M → M is annihilated by an element of theform (1 + z) where z ∈ m. As O is a local ring, the element (1 + z) must be a unit of O, thus M = 0 as


claim. The exactness of the Koszul complex at E follows immediately, as it is equivalent to the surjectivityof the morphism d2 : F → R. �

We end this appendix by proving its main result:

Proof of Theorem A.1. Since (inw(f1), · · · , inw(fs)) is regular in O, Lemma A.11 ensures that the Koszulcomplex (A.1) is exact at E. In turn, Proposition A.4 implies that (f1, . . . , fs) is a regular sequence in O.This proves item (1) of the statement.

To finish, we must show that the w-initial forms {inw(f1), · · · , inw(fs)} generate the w-initial idealinw(J)O. By definition, the ideal generated by these forms is contained in inw(J)O. As inw(J)O is gener-ated over O by all elements inw(g) with g ∈ J , the reverse inclusion will follow immediately if we show thatinw(g) ∈ (inw(f1), · · · , inw(fs))O. This identity is a direct consequence of Lemma A.8. Therefore, item (2)holds. This concludes our proof. �

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Authors’ addresses:

M.A. Cueto, Mathematics Department, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA.

Email address: [email protected]

P. Popescu-Pampu, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painleve, F-59000 Lille, France.


D. Stepanov, Laboratory of Algebraic Geometry and Homological Algebra, Department of Higher Mathematics,

Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow, 141701, Russia.


https://arxiv.org/abs/2202.00587

[email protected]

[email protected]

[email protected]

arXiv:2108.05912v2 [math.AG] 14 Mar 2022

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