Fitting ideals and multiple points of surface parameterizations Laurent Bus´ e Galaad, INRIA Sophia Antipolis [email protected]June 4, 2013 MEGA conference Joint work with M. Chardin (University of Paris VI) and N. Botbol (University of Buenos Aires).
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Fitting ideals and multiple points of surface ... · Fitting ideals and multiple points of surface parameterizations Laurent Bus e Galaad, INRIA Sophia Antipolis [email protected]
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Fitting ideals and multiple points of surfaceparameterizations
I Fitting image: subscheme of A2 defined by the initial Fitting ideal ofK[t] seen as a K[x , y ]-module (direct image):
K[x , y ]2
−y x2
x −y
−−−−−−−−−−−→ K[x , y ]2 → K[t]→ 0
Fitting ideals
Let A be a ring and M a A-module of finite presentation:
Aq ψ−→ Ap → M → 0.
Definition
The k th Fitting ideal of M is the ideal Fk(M) generated by all the p − kminors of ψ. It does not depend on the choice of ψ.
Properties
I F0(M) ⊂ F1(M) ⊂ · · · ⊂ A.
I annA(M)p ⊂ F0(M) ⊂ annA(M)
I A→ B a ring map, then
Fk(M ⊗A B) = Fk(M).B
(Fitting ideals commute with base change)
Fitting ideals
Let A be a ring and M a A-module of finite presentation:
Aq ψ−→ Ap → M → 0.
Definition
The k th Fitting ideal of M is the ideal Fk(M) generated by all the p − kminors of ψ. It does not depend on the choice of ψ.
Properties
I F0(M) ⊂ F1(M) ⊂ · · · ⊂ A.
I annA(M)p ⊂ F0(M) ⊂ annA(M)
I A→ B a ring map, then
Fk(M ⊗A B) = Fk(M).B
(Fitting ideals commute with base change)
Some related works
I If X is a scheme and M a coherent sheaf on X , on can define thesheaves of ideals Fk(M).
I Given f : X → Y a finite map, one can consider Fk(f∗OX ).
Tessier’s idea
Define the image of a finite map as the Fitting ideal F0(f∗OX ) instead ofannOY
(f∗OX ), in order to get stability under base change.
In this settings, works by Teissier (1976), by Mond, Pellikaan (1989),and also by Piene (1978) and (Kleiman, Lipman, Ulrich 1992) inrelation with the double point formula.
The case of parameterized plane curves
The multiple points sets ∆k(φ) can be described in terms of the Fittingideals of the associated classical implicitizing Sylvester matrix.
Some related works
I If X is a scheme and M a coherent sheaf on X , on can define thesheaves of ideals Fk(M).
I Given f : X → Y a finite map, one can consider Fk(f∗OX ).
Tessier’s idea
Define the image of a finite map as the Fitting ideal F0(f∗OX ) instead ofannOY
(f∗OX ), in order to get stability under base change.
In this settings, works by Teissier (1976), by Mond, Pellikaan (1989),and also by Piene (1978) and (Kleiman, Lipman, Ulrich 1992) inrelation with the double point formula.
The case of parameterized plane curves
The multiple points sets ∆k(φ) can be described in terms of the Fittingideals of the associated classical implicitizing Sylvester matrix.
Surface parameterizationsSupose given a birational surface parameterization
φ : P2 99K P3
(s0 : s1 : s2) 7→ (f0 : f1 : f2 : f3)(s0, s1, s2)
I Notation: S := K[s0, s1, s2], I = (f0, f1, f2, f3) ⊂ Sd , d ≥ 1.I Base points: B = Proj(S/I ) ⊂ P2, assumed finite (w.l.o.g.).
Let Γ be the closure of the graph of P2 \ B φ−→ P3:
Γ
π1
��
π2
##
� � //P2 × P3
P2
φ//P3
I Fiber at a point P ∈ P3:
π−12 (P) = Γ×P3 Spec(κ(P)) ⊂ P2κ(P)
where κ(P) is the residue field.
Surface parameterizationsSupose given a birational surface parameterization
φ : P2 99K P3
(s0 : s1 : s2) 7→ (f0 : f1 : f2 : f3)(s0, s1, s2)
I Notation: S := K[s0, s1, s2], I = (f0, f1, f2, f3) ⊂ Sd , d ≥ 1.I Base points: B = Proj(S/I ) ⊂ P2, assumed finite (w.l.o.g.).
Let Γ be the closure of the graph of P2 \ B φ−→ P3:
Γ
π1
��
π2
##
� � //P2 × P3
P2
φ//P3
I Fiber at a point P ∈ P3:
π−12 (P) = Γ×P3 Spec(κ(P)) ⊂ P2κ(P)
where κ(P) is the residue field.
Blow-up algebras
Algebraically, Γ = Proj(ReesS(I )), the blow up of φ along B.
Property
If B is a local complete intersection, then
Γ = Proj(ReesS(I )) = Proj(SymS(I ))
Compared to the Rees algebra, the symmetric algebra is a more friendlyto computations:
SymS(I ) = S [x0, x1, x2, x3]/( 3∑i=0
gixi :3∑
i=0
gi fi = 0
)
⇒ from now on, B is assumed to be a local complete intersection.
Blow-up algebras
Algebraically, Γ = Proj(ReesS(I )), the blow up of φ along B.
Property
If B is a local complete intersection, then
Γ = Proj(ReesS(I )) = Proj(SymS(I ))
Compared to the Rees algebra, the symmetric algebra is a more friendlyto computations:
SymS(I ) = S [x0, x1, x2, x3]/( 3∑i=0
gixi :3∑
i=0
gi fi = 0
)
⇒ from now on, B is assumed to be a local complete intersection.
Fitting ideals associated to φ
Notation : R := K[x0, x1, x2, x3], so that P3 = Proj(R)
Projecting Γ on P3 amounts to take graded parts of SymS(I )
Definition
For all integer ν ≥ 0, we have
(Z1)ν ⊂ ⊕3i=0R[s0, s1, s2]ν
M(φ)ν−−−−→ R[s0, s1, s2]ν → SymS(I )ν → 0
(g0, g1, g2, g3) :3∑
i=0
gi fi = 0 7→3∑
i=0
gixi
I get a family of matrices M(φ)ν indexed by ν ∈ NI define the associated Fitting ideals: