Constructing algebraic varieties via commutative algebra Miles Reid

Constructing algebraic varieties via

commutative algebra

Miles Reid (EAGER network)

Abstract Problems on the existence and moduli of

abstract varieties in the classification of varieties can

often be studied by embedding the variety X into

projective space, preferably in terms of an intrinsically

determined ample line bundle L such as the (anti-)

canonical class or its submultiples. A comparatively

modern twist on this old story is to study the graded

coordinate ring

R(X,L) =⊕n≥0

H0(X,L⊗n),

which in interesting cases is a Gorenstein ring; this

makes available theoretical and computations tools from

commutative algebra and computer algebra. The

varieties of interest are curves, surfaces, 3-folds, and

historical results of Enriques, Fano and others are

sometimes available to serve as a guide. This has been a

prominent area of work within European algebraic

geometry in recent decades, and the lecture will present

the current state of knowledge, together with some

recent examples.

1

EAGER

EAGER builds on the success of the former EU networks AGEand EuroProj that have run since the late 1980s. EAGER issupported by the programme Improving Human Potential andthe Socio-economic Knowledge base of the EuropeanCommission, Contract No. HPRN-CT-2000-00099.

(1) North Italy, A. Conte (Torino), overall coordinator

(2) Spain, J.-C. Naranjo (Barcelona)

(3) South Germany and Romania, Fabrizio Catanese(Bayreuth)

(4) North Germany, K. Hulek (Hannover)

(5) France, A. Hirschowitz (Nice)

(6) Scandinavia, K. Ranestad (Oslo)

(7) South Italy, C. Ciliberto (Roma)

(8) Israel, M. Teicher (Bar-Ilan)

(9) Benelux, E. Looijenga (Utrecht)

(10) Poland, J. Wisniewski (Warszawa)

(11) United Kingdom, Portugal and Hungary, Miles Reid(Warwick) → link to independent group “Vectorbundles on algebraic curves”(VBAC)

(12) Switzerland, Christian Okonek (Zuerich)

(13) Program Management node, W. Decker

2

EAGER objectives

(1) Classification of algebraic varieties

(2) Homological and categorical methods

(3) Moduli stacks of curves

(4) Moduli of vector bundles

(5) Abelian varieties and their moduli

(6) Hodge theory and algebraic cycles

(7) Toric methods and group actions

(8) Computer algebra

(9) Coding theory

(10) Computer Aided Geometric Design

Other Calabi–Yau manifolds and mirror symmetry.Topology of algebraic surfaces and 4-manifolds. Moduli spaces.Algebraic stacks and their Gromov–Witten invariants. Freeresolutions, homological algebra and derived categories.Birational methods. Deformation theory. Analytic anddifferential geometric methods. Syzygies and homologicalmethods, derived categories.

Today’s lecture only treats a small fraction of the firsttopic, namely:

Classification of algebraic varieties via commutativealgebra methods.

3

Classification of varieties

The classification of surfaces goes back to the 19th century.

1846 Cayley and Salmon: 27 lines on S3 ⊂ P3

1860s Riemann surfaces, Brill–Noether, RR theorem

1890–1910 Castelnuovo, Enriques and other: Birationalclassification of surfaces by their plurigenera

1930s Enriques and students: Surfaces of general type

1930s Fano: 3-folds V2g−2 ⊂ Pg+1

1950s Kodaira: deformation theory, moduli, classification ofcomplex analytic surfaces

1980s Mori theory, minimal models of 3-folds. The conclusionthat classification is the division K < 0, K = 0, K > 0plus fibrations, where K is the canonical class.

1980s Differentiable and symplectic 4-manifolds (Donaldsonand others)

1990s Calabi–Yau 3-folds, orbifolds, mirror symmetry.

EAGERists are involved in all these topics (and many more, ofcourse).

Any number of survey lectures could be made out of otherEAGER topics.

4

Preliminary philosophical remarks

Surfaces In what follows the ultimate aim (not necessarilyexpressed) is the study of regular surfaces of general type, forexample, the simply connected Godeaux surfaces (that is,canonical surfaces S with pg = 0, K2

S = 1). This is a maturesubject, that involves most other areas of geometry. To studyS, it may be convenient to know a lot about curves C ⊂ S,possibly passing through singular points of S; or it may beconvenient to express S as a hypersurface section of somehigher dimensional “key variety”, e.g., a Fano 3-fold or Fano4-fold, possibly with orbifold singularities. Surprisingly, itturns out to be advantageous in some problems not to worrytoo much in advance what dimension of variety we arestudying: taking a hypersurface section is a known operation.

Commutative algebra The geometric constructions ofEnriques, Horikawa and others can often be interpreted inalgebraic terms as constructions of rings by generators andrelations. As samples:

(1) The hypersurface Xd ⊂ Pn defined by fd = 0 hashomogeneous coordinate ring the graded ringC[x0, x1, . . . , xn]/(fd).

(2) The geometric idea of projection corresponds algebraicallyto elimination of variables.

(3) “Key varieties” may have a homological or commutativealgebra treatment, such as determinantal form of theequations.

5

Definition of graded ring

A graded ring R =⊕n≥0Rn is a (commutative) ring with a

grading such that multiplication does Ri ×Rj → Ri+j.

Extra assumptions The following are often in force:

(1) R0 = k is a field (often k = C);

(2) The maximal ideal m =⊕n>0Rn is finitely generated.

=⇒ R = k[x0, . . . , xn]/IR,

where the generators xi ∈ Rai of m have wtxi = ai, andIR is the homogeneous ideal of relations.

(3) R is an integral domain.

Example The standard textbooks define a projective varietyto be a closed subvariety X ⊂ Pn in “straight” projectivespace Pn (all the generators of degree 1, so xi ∈ R1). Write

IX =⊕d≥0{forms of degree d vanishing on X}.

Then IX is a homogeneous ideal and k[X] = k[x0, . . . , xn]/IX isthe coordinate ring of X. Here R is generated by its elementsof degree 1; we are usually interested in the more general caseof varieties in weighted projective space.

For details, see my website + algebraic geometry links +surfaces + graded rings and homework.

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The Proj construction R 7→ ProjR

As described in [EGA2] or [Hartshorne, Chap. II] or my notes(webloc. cit.), X = ProjR is defined as the quotient(SpecR \ 0)/C∗ of the variety SpecR = V (I) ⊂ Cn+1 by theaction of the multiplicative group C∗ = Gm(k) induced by thegrading.

In more detail, if R = k[x0, . . . , xn]/I with wt xi = ai thenλ ∈ C∗ acts on R by multiplication by λn on Rn, that is,

λ:xi → λaixi.

It therefore acts on the affine variety

SpecR = V (I) ⊂ Cn+1.

Note the philosophy: grading = C∗ action.

The origin 0 ∈ Cn+1 is in the closure of every orbit (because(0, 0, . . . , 0) = limλ→0(λ

a0x0, . . . , λanxn); this uses the fact that

the grading of R =⊕Rn is by N with n > 0, or wtxi = ai > 0.

Therefore we must exclude the unstable point 0 to be able totake a sensible quotient.

For all f ∈ Rd homogeneous of degree d > 0, form the ring(R

[1

f

])0=

{g

f e

∣∣∣∣∣ wt g = de

}⊂ FracR (1)

consisting of rational functions that are homogeneous of deg 0with only f or its powers in the denominator. Then define

Xf = Spec

(R

[1

f

])0, and X =

⋃f∈Rd

Xf .

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In other words, on taking the quotient (SpecR \ 0)/C∗:

(1) The typical C∗ invariant open set is (f 6= 0) for f ∈ Rd.

(2) the ring (1) is the ring of all C∗-invariant regularfunctions on this open.

Thus the quotient ProjR is the space of orbits of the C∗

action, with all C∗-invariant functions.

Remark X = ProjR is really a stack, and it is sometimesconvenient to treat it as an orbifold. It is a projective schemeX,OX , but it has the extra structure of the sheaves OX(k) forall k ∈ Z, defined by

Γ(Xf ,OX(k)) =

{g

f e

∣∣∣∣∣ wt g = de+ k

}⊂ FracR.

Then⊕k∈ZOX(k) is a sheaf of graded algebras.

For straight projective space (that is, wtxi = 1 for all xi),OX(1) is an ample invertible sheaf, and

OX(k) = OX(1)⊗k.

But for wP we must take⊕OX(k) as extra data. For example,

if all the ai have some common factor q | ai then Rn = 0 for alln not divisible by q, and so OX(k) = 0. In this case we saythat X has nontrivial orbifold structure in codim 0.

Examples C2g+2 ⊂ P(1, 1, g + 1) defined byy2 = f2g+2(x1, x2) is a hyperelliptic curve of genus g.X10 ⊂ P(1, 1, 2, 5) defined by z2 = f10(x1, x2, y) is a famousexample of Enriques and Kodaira of a canonical surface withpg = 2, K2 = 1.

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Hilbert series

It follows from my assumptions on R that Rn if a finitedimensional vector space over R0 = C for each n. Set

Pn(R) = dimkRn and PR(t) =∑n≥0

Pntn.

The formal power series PR(t) is the Hilbert series of R. Underour assumptions it is a rational function in t; thus

R = k[x0, . . . , xn]/IR with wt xi = ai

implies that∏ni=0(1− tai) · PR(t) is a polynomial in t, called the

Hilbert numerator; it contains information and hints as to thehomological algebra or commutative algebra properties of R.

Examples

(1) If R = k[x0, . . . , xn] is the weighted polynomial ring then

PR(t) =1∏n

i=0(1− tai).

(2) If R = k[x0, . . . , xn]/(fd) is the ring of a weightedhypersurface of degree d in P(a0, . . . , an) then

PR(t) =1− td∏n

i=0(1− tai).

Likewise, a codim 2 complete intersection has Hilbertnumerator (1− td1)(1− td2).

See the homework sheet on webloc. cit. for more examples.

9

Hilbert series from orbifold RR

From now on, X is a projective variety, and OX(k) = OX(kA)with A an ample Q-divisor. So rA is an ample Cartier divisorfor some r > 0. Assume that

R = R(X,A) =⊕k≥0

H0(X,OX(kA)).

(This is an extra assumption on R, akin to projectivenormality.)

Usually the terms of the Hilbert series

Pn(R) = h0(X,OX(nA))

are given by RR and vanishing for n� 0, plus initialassumptions for small n. If A is Q-Cartier, the form of RR weneed is orbifold RR (also known as equivariant RR or theAtiyah–Singer Lefschetz formula). See [YPG, Chap. III] fordetails. A simple example gives the flavour.

Example C a curve, A = D+ arP with D an integral divisor,

r > 1 and a ∈ [1, . . . , r − 1] coprime to r. ThenOC(nA) = OC([nA]), where we round down the divisor nA tothe nearest integer (because a meromorphic function has polesof integral order), so that RR takes the form

χ(C,OC(nA)) = χ(OC([nA])) = 1− g + n degA−{na

r

}.

Here the fractional part{nar

}is the small change we lose on

rounding down nA to [nA].

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This introduces the orbifold correction term

− 1

1− tr·r−1∑i=1

{ia

r

}ti (2)

into the Hilbert series. (The effect of multiplying by1

1−tr = 1 + tr + t2r + · · · is just to repeat the rounding-downerrors periodically.)

Remark Set ab ≡ 1 mod r and let ε be a primitive rth rootof 1 (for example, ε = exp(2πi/r)). Then one checks that

1

1− εb=

r−1∑i=1

{ia

r

}εi

Thus the term (2) is “cyclotomic” in nature. Generalisations ofthis idea give very quick and convenient ways of calculating theorbifold contributions to RR. We are in fact close to the proofof the Atiyah–Singer equivariant Lefschetz formula: thedenominator is the equivariant Todd class det(ε : TX,P ). See[YPG, Chap. III].

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Example [Bauer, Catanese, Pignatelli] C a curve of genusg = 3 with points P,Q ∈ C such that P + 3Q = KC . Forexample, C = C4 ⊂ P2, with Q a flex and P the 4th point ofintersection of the flex line with C.

I choose the divisor A = 12P +Q. Then

h0(nA) =

1 n = 0;

1 n = 1;

2 n = 2 (P + 2Q = KC −Q is a g13);

3 n = 3 (3A = KC + 12P and g = 3);

−2 + 3n2 if n ≥ 4 even;

−2 + 3n−12 if n ≥ 4 odd.

Therefore

PC,A(t) = 1 + t+ 2t2 + 3t3 + 4t4 + 5t5 + 7t6 + · · ·(1− t2)PC,A(t) = 1 + t+ t2 + 2t3 + 2t4 + 2t5 + 3t6 +

∑3tn

(1− t)(1− t2)PC,A(t) = 1 + t3 + t6

Thus

PC,A(t) =1− t9

(1− t)(1− t2)(1− t3).

This gives C9 ⊂ P(1, 2, 3) as a possible model for C. Onechecks that it works: C has a 1

2(1) orbifold singular points at(0, 1, 0). The linear system |2A| = P + 2Q is the g1

3. R(C,A) isa Gorenstein ring because 3A = KC + 1

2P is the orbifoldcanonical class of C.

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Some classes of varieties to study

Regular surfaces of general type (Enriques) Assume thatKS is ample, and that q = h1(S,OS) = 0. (We say that S is aregular surface; irregular surfaces with q > 0 are studied bydifferent methods.)

Pn(S) =

1 k = 0;

pg k = 1 (the definition of pg);

1 + pg +

n2

K2 k ≥ 2 (by RR and vanishing).

An easy calculation gives

pS(t) =1 + (pg − 3)t+ (K2 − 2pg + 4)t2 + (pg − 3)t3 + t4

(1− t)3 .

About a dozen important cases were treated geometrically byEnriques, Kodaira, Horikawa and others. Algebraic treatmentby Ciliberto, Catanese, Reid and others.

Examples pg = 4, K2 = 6. The first possible case suggestedby the Hilbert series is S3,4 ⊂ P(1, 1, 1, 1, 2). This really works.There are lots of degenerate cases studied by Horikawa, andrecently by [Bauer, Catanese and Pignatelli]; see below. Thesituation for pg = 3, K2 = 2, 3, 4 or for pg = 2, K2 = 1, 2, 3 issimilar. Beyond these initial cases, the calculations get verydifficult.

Fano 3-folds Nonsingular 3-folds V with −KV ample,usually anticanonically embedded as V2g−2 ⊂ Pg+1. These werestudied by Fano in the 1930s and Iskovskikh from 1970s, laterMori and Mukai.

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Q-Fano 3-folds 3-folds V with terminal singularities and−KV ample (Mori, Reid and others, 1990s). In studying3-folds, terminal singularities are unavoidable; the mostimportant and interesting singularities are the cyclic quotientsingularities 1

r(1, a, r − a) with r ≥ 2 and a ∈ [1, r − 1] coprimeto r. Several hundred families of Q-Fano 3-folds are known, forexample the “famous 95” Fano hypersurfaces studied in [Corti,Pukhlikov, Reid]. See [DB].

Q-K3s These are surfaces X with quotient singularities andKX = OX , H1(OX) = 0 polarised by a Q-divisor. They appearnaturally as anticanonical surfaces X ∈ |−KV | on a Q-Fano3-fold V .

Remark It can happen that a surface of general type S iscontained in a Q-Fano 3-fold V , for example:

(1) S ∈ |−2KV |, so adjunction gives KS = KV |S;

(2) or V is a Q-Fano 3-fold of index 2 with −KV = 2A andS ∈ |3A|, so that KS = A|S.

A striking fact: the basket of singularities of V (giving thefractional contributions to its Hilbert series) is then alreadydetermined by S: in the two cases above

(1) V has basket (K2 − 4pg + 12)× 12(1, 1, 1). So for example,

if S has pg = 1, K2 = 1 then V has 9× 12(1, 1, 1) points,

whereas if S has pg = 1, K2 = 2 then V has 10× 12(1, 1, 1)

points. We really meet these cases below.

(2) V has basket (K2 − 3pg + 6)× 13(1, 2, 2).

This follows automatically from orbifold RR!

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Appendix: Cohen–Macaulay and Gorenstein

I omit the definitions and treatment by homological algebra,which are standard and not very difficult. In practice, we wantR to be Cohen–Macaulay and (better) Gorenstein; otherwisethe ring and the variety are very difficult to construct.

Criterion Let R = R(X,A). Then

R is Cohen–Macaulay if and only if H i(X,OX(kA)) = 0 forall i with 0 < i < dimX and all k, for i = 0 and k < 0,and for i = dimX and k � 0.

R is Gorenstein if and only if it is Cohen–Macaulay andKX = kA for some k ∈ Z.

Examples These conditions hold in most of our cases:

(1) X is a K3 surface with quotient singularities and A anample Weil divisor;

(2) X is a regular surface of general type and A = KX . ThenH1(KX) = 0 follows from regularity and Serre duality, andH1(nKX) = 0 for n ≥ 2 from Kodaira vanishing;

(3) V is a Q-Fano 3-fold of Fano index f and −KV = fA;

(4) C is an orbifold curve (with a point 1rP ), and we interpret

KC in the criterion as orbi-KC = KC + r−1r P .

The cone over a projectively embedded Abelian surfaces is asimple example of a geometrically interesting variety that isnot Cohen–Macaulay.

15

Application 1

Horikawa’s study of surfaces with pg = 4, K2 = 6 divides theminto several cases, and solves many problems, but leaves theexistence of degenerations between cases II and IIIb as an openquestion. [Bauer, Catanese, Pignatelli] have recently provedthat such a degeneration does occur.

II The case assumption is that |KX | is a free linear systemand defines a 3-to-1 morphism ϕKX

:X → Q ⊂ P3, where Q isthe quadric cone x1x3 = x2

2. In this case pulling back the pencilof the quadric cone provides a pencil |A| on the canonicalmodel X with 2A = KX . In general X has an orbifold point oftype 1

2(1, 1) over the vertex of Q. Restricting A to a generalC ∈ |A| gives rise to the example treated above of a curve ofgenus 3 and an orbifold divisor A = 1

2P +Q, so that2A = P + 2Q is a g1

3.

It follows that X = X9 ⊂ P(1, 1, 2, 3). This has all the requiredproperties, and every surface in II is given by this construction.

IIIb The case assumption is that |KX | has a double point asits base locus on the canonical model (or a −2-curve as basecomponent on the minimal model), and ϕK : X → Q ⊂ P3 is a2-to-1 morphism to the quadric cone. Then again KX = 2Awith A2 = 3/2. At the level of a general curve C ∈ |A|, thecurve C is a nonsingular hyperelliptic curve of genus 3, and therestriction A|C is 3

2P , where P is a Weierstrass point. (Thus2A = P + g1

2 can be viewed as a g13 with a fixed point.)

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[BCP2] (and also [Coughlan]) calculate R(C,A) and R(X,KX)in case IIIb:

R

(C,

1

2P

)= k[a, b, c]/(c2 − f7(a

4, b)) with wt a, b, c = 1, 4, 14,

giving C = C28 ⊂ P(1, 4, 14). Then R(C,A) = R(C, 32P ) is the

third Veronese embedding: it needs generators

x = a3, y = a2b, z = ab2, t = b3, u = ac, v = bc

with wt x, y, z, t, u, v = 1, 2, 3, 4, 5, 6. And relations

rank

x y z u

y z t v

≤ 1 (3)

(meaning the 2× 2 minors = 0, which gives 6 equations); and 3further equations derived from c2 = f7, of the form

u2 = [a2f ], uv = [abf ], v2 = [b2f ],

where [a2f ] means that we write out the termsa30, a26b, . . . , a2b7 of a2f in terms of x, y, z, t. If we grouptogether the terms in f as

f = a28 + a24b+ · · ·+ a4b6 + b7 = aA+ b4B

withA = A9, B = B4 ∈ k[x, y, z, t]

then the 3 final equations become

u2 = xA+ z2B, uv = yA+ ztB, v2 = zA+ t2B. (4)

This is the “rolling factors” format of [Dicks]: you go from onerelation to the next by replacing an entry in the top row of thematrix of (3) by an entry in the bottom.

17

(3–4) are 9 equations with 16 syzygies defining a codim 4Gorenstein ring. They can be written as the 4× 4 Pfaffians ofthe following extrasymmetric matrix:

M =

0 z x y u

t y z v

u v A

0 Bz

−sym Bt

of weights M =

0 3 1 2 54 2 3 6

5 6 94 7

8

The matrix M is skew, with the following extra symmetry: thetop right 3× 3 block is symmetric, and the bottom right 3× 3block is B times the top left. Thus instead of 15 independententries it only has 9, and likewise, only 9 independent 4× 4Pfaffians. The format relates closely to the Segre embedding ofP2 ×P2 as a (nongeneric) linear section of Grass(2, 6).

This format is flexible: it carries its own syzygies with it, sothat we can vary the entries as we like and obtain a flatdeformation. Replacing by

M =

λ z x y u

t y z v

u v A

Bλ Bz

−sym Bt

with a constant λ 6= 0 deforms the hyperelliptic curve to anonhyperelliptic trigonal curve. Similarly (but with some morework), one can prove that the surfaces in case IIIb have smalldeformations in II.

18

Appendix: All about Pfaffians

Let M0 = {mij} be a 2k × 2k skew matrix. Its Pfaffian is

Pf M0 =∑′

sign(σ)k∏i=1

mσ(2i−1)σ(2i);

(sum over the symmetric group S2k), and∑′ means that we

only take 1 occurrence of each repeated factor. Skewsymmetrycauses each term to occur 2k · k! times, so the Pfaffian consistsof

2k!

2k · k!= 1 · 3 · · · (2k − 1)

terms. For example, a 4× 4 Pfaffian is of the form

Pf12.34 = m12m34 −m13m24 +m14m23

which is familiar as the Plucker equations of Grass(2, n).

In fact detM0 = (Pf M0)2. The Pfaffian is a skew determinant,

and every aspect of the theory of determinants extends toPfaffians. For example, it follows from the definition that aPfaffian can be expanded along any row exactly like adeterminant: thus a 6× 6 Pfaffian is

Pf12.34.56 = m12 · Pf34.56−m13 · Pf24.56 + · · · .If M is a (2k + 1)× (2k + 1) skew matrix, write

Pfi = (−1)i Pf Mi,

where Mi is the skew 2k × 2k matrix obtained by deleting theith row and column from M . Then the adjoint matrix of M(matrix of 2k × 2k cofactors) is the matrix of rank 1 (or 0)

adjM = Pf ·t Pf, where Pf = (Pf1, . . . ,Pf2k+1)

Since detM = 0 we get Pf ·M = 0, and if M has rank 2k thenPf generates kerM (skew Cramer’s rule).

19

Application 2

Surfaces with pg = 1, K2 = 2 were studied in [Catanese andDebarre], following Enriques; an alternative construction as asection of a higher dimensional variety was given by JanStevens in 1995 (but as far as I know not written down).

I start from the graded ring over the canonical curve C ∈ |KS|:a reasonably general 4× 4 symmetric matrix M of linear formson P2

y1,y2,y3defines an invertible sheaf OC(A) on the plane

quartic C = C4 : (detM = 0) ⊂ P2, with the resolution

OC(A)← 4OP2(−1)M←− 4OP2(−2)← 0, (5)

and satisfying OC(2A) = KC (in other words, A is anineffective theta characteristic on C). The correspondinggraded ring

R(C,A) = k[y1, y2, y3, z1, z2, z3, z4]/IC

is generated by y1, y2, y3 ∈ H0(OC(2A)) andz1, . . . , z4 ∈ H0(OC(3A)) = OC(A)(1) with relations(z1, . . . , z4)M = 0 from (5) and zizj = Mij (the ijth maximalminor of M . These equations define a codim 5 embeddingC ⊂ P(23, 34) with Hilbert numerator

1− 4t5 − 10t6 + 15t8 + 20t9 − 20t11 − · · ·

The same construction starting from a 4× 4 symmetric matrixM over P3 leads to a quartic K3 surface X4 ⊂ P3 carrying anineffective Weil divisor AX with a resolution similar to (5), andR(X,A) embeds X into P(24, 34). However, now X has 10nodes at points where rankM = 2. These are 1

2(1, 1) orbifoldpoints at which OX(AX) is the odd eigensheaf.

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The problem is to deform the graded ring R(C,A) orR(X,AX) with new generators of degree 1. First project Xfrom a chosen node to X ′6,6 ⊂ P(2, 2, 2, 3, 3); the exceptionalcurve of this projection is P1 = P(1, 1) embedded intoP(2, 2, 2, 3, 3). Since P(2, 2, 2, 3, 3) has no forms of degree 1,this embedding is not projectively normal ; in coordinates it is

(v, w) 7→ (v2, vw, w2, v3 + αv2w, βvw2 + w3)

with 1 + αβ 6= 0.

The following result is joint work with Grzegorz Kapustka andMichal Kapustka (who held an EAGER visiting studentship atWarwick in spring 2004).

Claim General forms of degree 1, 2, 2, 2, 3, 3 define anembedding P2 ∼= Π ⊂ P(1, 2, 2, 2, 3, 3) with image Π containedin 3 sextics. The complete intersection of two general sexticsthrough Π is a Q-Fano 3-fold V ′6,6 with 9× 1

2(1, 1, 1) orbifoldpoints on P2

y1,y2,y3, 24 ordinary nodes on Π, and nonsingular

otherwise.

The 24 nodes of V ′6,6 on Π are resolved by the (small) blowupV ′′ → V ′6,6 of Π, and the birational image E ⊂ V ′′ of Π hasE ∼= P2, OE(−E) ∼= OP2(2); it contracts to a tenth orbifoldpoint 1

2(1, 1, 1) on a Fano V ⊂ P(1, 24, 34).

The proof is a calculation in computer algebra. According toresults of Jan Stevens, V actually extends to a Fano 6-foldW ⊂ P(14, 24, 34) of Fano index 4 having 10 isolated orbifoldpoints of type 1

2(1, . . . , 1). (It can be obtained by an immersionP5 → P(14, 23, 32) contained in two sextics, but thecomputation is quite bulky.)

21

References

[BCP1] Ingrid Bauer, Fabrizio Catanese and RobertoPignatelli, Canonical rings of surfaces whose canonicalsystem has base points, in Complex geometry(Gottingen, 2000), Springer, Berlin, 2002, pp. 37–72,

[BCP2] Ingrid Bauer, Fabrizio Catanese and RobertoPignatelli, The moduli space of surfaces withK2 = 6, pg = 4, math.AG/0408062, 17 pp.

[DB] Gavin Brown, Graded rings database, online athttp://www.maths.warwick.ac.uk/ gavinb/grdb.html

[CD] F. Catanese and O. Debarre, Surfaces with K2 = 2,pg = 1, q = 0, J. reine angew. Math. 395 (1989) 1–55

[CPR] A. Corti, A. Pukhlikov and M. Reid, Birationally rigidFano hypersurfaces, in Explicit birational geometry of3-folds, A. Corti and M. Reid (eds.), CUP 2000,pp. 175–258

[Coughlan] S. Coughlan, Univ. of Warwick MSc dissertation(in preparation), Sep 2004

[Dicks] M. Reid, Surfaces with pg = 3, K2 = 4 according to E.Horikawa and D. Dicks, in Proceedings of Algebraicgeometry mini-symposium (Tokyo Univ., Dec 1989,distributed in Japan only), 1–22 (get from my website)

[YPG] M. Reid, Young person’s guide to canonicalsingularities, in Algebraic Geometry, Bowdoin 1985, ed.S. Bloch, Proc. of Symposia in Pure Math. 46, A.M.S.(1987), vol. 1, pp. 345–414

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