An introduction to the Cayley-Bacharach theorems following Eisenbud, Green, Harris, Cayley-Bacharach theorems and conjectures
Remi´ Bignalet
May 17, 2018
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 1 / 1 r12 r13 r23
Pappus’s Theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 2 / 1 r13 r23
Pappus’s Theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 2 / 1 r23
Pappus’s Theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 2 / 1 Pappus’s Theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 2 / 1 Pappus’s Theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 2 / 1 Pappus’s Theorem
Theorem (First version of the Cayley-Bacharach theorem, IVth century AC)
Lest L and M be two lines in the plane. Lest p1, p2 and p3 be distinct points of L and let q1, q2 and q3 be distinct points on M all distinct from the point L ∩ M. If for each j 6= l ∈ {1,2,3} we let rjk be the point of intersection of the lines pj qk and pk qj , then the three points rjk are colinear.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 3 / 1 r12 r13
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Pascal’s theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 4 / 1 r13
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Pascal’s theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 4 / 1 r23
Pascal’s theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 4 / 1 Pascal’s theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 4 / 1 Pascal’s theorem
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 4 / 1 r12 r13
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Pascal’s Theorem
Theorem (Pascal’s theorem, 1640) If a hexagon is inscribed in a conic in the projective plane, the opposite sides of the hexagon meet in three collinear points.
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 5 / 1 Pascal’s Theorem
Theorem (Pascal’s theorem, 1640) If a hexagon is inscribed in a conic in the projective plane, the opposite sides of the hexagon meet in three collinear points.
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 5 / 1 Homogeneous coordinates: (x : y : z) = (λx : λy : λz)
Remark i j k Given f = ∑ αijk x y z homogeneous polynomial of degree i+j+k=d d in three variables:
2 {(x0 : y0 : z0) ∈ P (C),f (x0,y0,z0) = 0}
makes sense as a subset of P2(C).
Projective Geometry
Definition
2 3 P (C) = C \{0}/(x,y,z) ∼ (λx,λy,λz)
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 6 / 1 Remark i j k Given f = ∑ αijk x y z homogeneous polynomial of degree i+j+k=d d in three variables:
2 {(x0 : y0 : z0) ∈ P (C),f (x0,y0,z0) = 0}
makes sense as a subset of P2(C).
Projective Geometry
Definition
2 3 P (C) = C \{0}/(x,y,z) ∼ (λx,λy,λz) Homogeneous coordinates: (x : y : z) = (λx : λy : λz)
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 6 / 1 homogeneous polynomial of degree
d in three variables:
2 {(x0 : y0 : z0) ∈ P (C),f (x0,y0,z0) = 0}
makes sense as a subset of P2(C).
Projective Geometry
Definition
2 3 P (C) = C \{0}/(x,y,z) ∼ (λx,λy,λz) Homogeneous coordinates: (x : y : z) = (λx : λy : λz)
Remark i j k Given f = ∑ αijk x y z i+j+k=d
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 6 / 1 Projective Geometry
Definition
2 3 P (C) = C \{0}/(x,y,z) ∼ (λx,λy,λz) Homogeneous coordinates: (x : y : z) = (λx : λy : λz)
Remark i j k Given f = ∑ αijk x y z homogeneous polynomial of degree i+j+k=d d in three variables:
2 {(x0 : y0 : z0) ∈ P (C),f (x0,y0,z0) = 0}
makes sense as a subset of P2(C).
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 6 / 1 1 2 { 2 y + xz = 0}
{xz = 0}
1 2 { 8 y + xz = 0}
1 2 { 4 y + xz = 0}
Pappus’ theorem as deformation
{y 2 + xz = 0}
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 7 / 1 {xz = 0}
1 2 { 8 y + xz = 0}
1 2 { 4 y + xz = 0}
Pappus’ theorem as deformation
2 1 2 {y + xz = 0} { 2 y + xz = 0}
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 7 / 1 {xz = 0}
1 2 { 8 y + xz = 0}
Pappus’ theorem as deformation
2 1 2 {y + xz = 0} { 2 y + xz = 0}
1 2 { 4 y + xz = 0}
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 7 / 1 {xz = 0}
Pappus’ theorem as deformation
2 1 2 {y + xz = 0} { 2 y + xz = 0}
1 2 { 8 y + xz = 0}
1 2 { 4 y + xz = 0}
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 7 / 1 Pappus’ theorem as deformation
2 1 2 {y + xz = 0} { 2 y + xz = 0}
{xz = 0}
1 2 { 8 y + xz = 0}
1 2 { 4 y + xz = 0}
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 7 / 1 t p1 1 t t t t det(r12,r13,r23) = 0 ptpt 2 12 13 23 pp222 pt p3
tt rr t rr121212 12 t t t r13rr23t rr232323
qt2t qt qq22t q3 q2 qt q1
Pappus’ theorem as deformation
pt 1 pt t t t 2 det(r12,r13,r23) = 0
t p3
t r12 r t t 13 r23
t q3 qt t 2 q1
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 8 / 1 pt p1 t p det(r t ,r t ,r t ) = 0 ptt 2 12 13 23 pp222 pt p3
rrtt r1212t12 r12 t t t r13rr23t rr232323
qt2t qt qq22 q3 qt qt 2 q1
Pappus’ theorem as deformation
t p1 t det(r t ,r t ,r t ) = 0 p2 12 13 23
t p3
r t 12 r t t 13 r23
t t q3 q2 t q1
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 8 / 1 t p1 1 t t t t p2t det(r12,r13,r23) = 0 tp2 12 13 23 pp2 2 pt p3
t r tt rrr1212 t 12 r13rrt t 13rr232323
t t qq2 q3 q2t 3 qt2 qt 2 q1
Pappus’ theorem as deformation
pt 1 t t t t det(r12,r13,r23) = 0 p2
t p3
r t 12 t r13 t r23
t t q2 q3 t q1
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 8 / 1 t p1 1 t t t t p2t det(r12,r13,r23) = 0 ppt 2 12 13 23 p22 pt p3
t r t rr12t12 r12 t t r13rr23t r2323
t t q2 q3 q2t 3 qt2 qt 2 q1
Pappus’ theorem as deformation
pt 1 t t t t det(r12,r13,r23) = 0 p2 t p3
t r12 r t t 13r23
t t q2 q3 t q1
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 8 / 1 pt 1 t t t t p2t det(r ,r ,r ) = 0 tpt 2 12 13 23 pp22 t p3
tt rrr1212tt r12 t t t r13 rt rr232323
qqtt qt q22t 3 qt2 t 2 q1
Pappus’ theorem as deformation
p1 det(r12,r13,r23) = 0 p2
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 8 / 1 r12
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A first limit case
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 9 / 1 r12
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A first limit case
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 9 / 1 r12
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A first limit case
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 9 / 1 A first limit case
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 9 / 1 A first limit case
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 9 / 1 Chasles’ Theorem
Theorem (Chasles’ Theorem) n Let X1 and X2 ⊂ P (C) be two cubic plane curve meeting in n nine points p1,...,p9. If X ⊂ P (C) is any cubic containing a priori p1,...,p8, then X contains p9 as well.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 10 / 1 X1
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Application (Pascal’s theorem)
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 11 / 1 X2
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Application (Pascal’s theorem)
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 11 / 1 X1
Application
Application (Pascal’s theorem)
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 11 / 1 Application
Application (Pascal’s theorem)
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Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 11 / 1 The number of condition imposed by Γ on polynomials of degree d is denoted by hΓ(d).
Example The set Γ of 3 collinear points imposes two conditions on polynomials of degree 1 i.e. hΓ(1) = 2.
Terminology
Definition If Γ ⊂ P2(C) is a set of distinct points, we say that Γ imposes l conditions on the polynomial of degree d if the subspace h C[x,y,z]d vanishing at p1,...,pm has codimension l.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 12 / 1 Example The set Γ of 3 collinear points imposes two conditions on polynomials of degree 1 i.e. hΓ(1) = 2.
Terminology
Definition If Γ ⊂ P2(C) is a set of distinct points, we say that Γ imposes l conditions on the polynomial of degree d if the subspace h C[x,y,z]d vanishing at p1,...,pm has codimension l.
The number of condition imposed by Γ on polynomials of degree d is denoted by hΓ(d).
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 12 / 1 Terminology
Definition If Γ ⊂ P2(C) is a set of distinct points, we say that Γ imposes l conditions on the polynomial of degree d if the subspace h C[x,y,z]d vanishing at p1,...,pm has codimension l.
The number of condition imposed by Γ on polynomials of degree d is denoted by hΓ(d).
Example The set Γ of 3 collinear points imposes two conditions on polynomials of degree 1 i.e. hΓ(1) = 2.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 12 / 1 0 then for all Γ = {p1,...,p8},
hΓ(3) = hΓ0 (3)
Remark
Proof is actually showing that hΓ(3) = hΓ0 (3) = 8.
Chasles’ theorem
Theorem
Given Γ = {p1,...,p9} = X1 ∩ X2 where X1 and X2 are plane cubics,
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 13 / 1 hΓ(3) = hΓ0 (3)
Remark
Proof is actually showing that hΓ(3) = hΓ0 (3) = 8.
Chasles’ theorem
Theorem
Given Γ = {p1,...,p9} = X1 ∩ X2 where X1 and X2 are plane 0 cubics, then for all Γ = {p1,...,p8},
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 13 / 1 Remark
Proof is actually showing that hΓ(3) = hΓ0 (3) = 8.
Chasles’ theorem
Theorem
Given Γ = {p1,...,p9} = X1 ∩ X2 where X1 and X2 are plane 0 cubics, then for all Γ = {p1,...,p8},
hΓ(3) = hΓ0 (3)
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 13 / 1 Chasles’ theorem
Theorem
Given Γ = {p1,...,p9} = X1 ∩ X2 where X1 and X2 are plane 0 cubics, then for all Γ = {p1,...,p8},
hΓ(3) = hΓ0 (3)
Remark
Proof is actually showing that hΓ(3) = hΓ0 (3) = 8.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 13 / 1 Ω fail to impose independent conditions on curves of degree d if and only if either d + 2 points of Ω are collinear or n = 2d + 2 and Ω is contained in a conic.
A lemma
Lemma 2 Let Ω = {p1,...,pn} ⊂ P be a set of n distinct points and let an integer d such that n ≤ 2d + 2.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 14 / 1 A lemma
Lemma 2 Let Ω = {p1,...,pn} ⊂ P be a set of n distinct points and let an integer d such that n ≤ 2d + 2.
Ω fail to impose independent conditions on curves of degree d if and only if either d + 2 points of Ω are collinear or n = 2d + 2 and Ω is contained in a conic.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 14 / 1 Example
y = x2 y = x2 y = x2
y = 1 y = −1 y = 0
One idea
Theorem (Bezout’s´ theorem) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively. If X1 and X2 have no common component then they meet in d × e points.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 15 / 1 y = x2 y = x2
y = −1 y = 0
One idea
Theorem (Bezout’s´ theorem) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively. If X1 and X2 have no common component then they meet in d × e points.
Example
y = x2
y = 1
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 15 / 1 y = x2
y = −1
One idea
Theorem (Bezout’s´ theorem) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively. If X1 and X2 have no common component then they meet in d × e points.
Example
y = x2 y = x2
y = 1
y = 0
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 15 / 1 One idea
Theorem (Bezout’s´ theorem) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively. If X1 and X2 have no common component then they meet in d × e points.
Example
y = x2 y = x2 y = x2
y = 1 y = −1 y = 0
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 15 / 1 The theory of curves (XIXth century)
Theorem (Cayley-Bacharach theorem, version 4) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively, meeting in a collection of d × e distinct points n Γ = X1 ∩ X2 = {p1,...,pde}. If C ⊂ P (C) is any plane curve of degree d + e − 3 containing all but one point of Γ, then C contains all of Γ.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 16 / 1 Set s = d + e − 3.
If k ≤ s is a nonnegative integer, then the dimension of the vector space of polynomials of degree k, vanishing on Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions on polynomials of degree s − k.
The theory of curves (XIXth century)
Theorem (Cayley-Bacharach theorem, version 5) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively, meeting in a collection of d × e distinct points Γ = X1 ∩ X2 = {p1,...,pde} and suppose that Γ is the disjoint union of susbsets Γ0 and Γ00.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 17 / 1 If k ≤ s is a nonnegative integer, then the dimension of the vector space of polynomials of degree k, vanishing on Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions on polynomials of degree s − k.
The theory of curves (XIXth century)
Theorem (Cayley-Bacharach theorem, version 5) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively, meeting in a collection of d × e distinct points Γ = X1 ∩ X2 = {p1,...,pde} and suppose that Γ is the disjoint union of susbsets Γ0 and Γ00. Set s = d + e − 3.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 17 / 1 The theory of curves (XIXth century)
Theorem (Cayley-Bacharach theorem, version 5) 2 Let X1 and X2 ⊂ P (C) be plane curves of degree d and e respectively, meeting in a collection of d × e distinct points Γ = X1 ∩ X2 = {p1,...,pde} and suppose that Γ is the disjoint union of susbsets Γ0 and Γ00. Set s = d + e − 3.
If k ≤ s is a nonnegative integer, then the dimension of the vector space of polynomials of degree k, vanishing on Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions on polynomials of degree s − k.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 17 / 1 and suppose that the 0 intersection Γ = X1 ∩ ... ∩ Xn is the disjoint union of subsets Γ 00 and Γ . Set s = ∑di − n − 1. If k ≤ s is a nonnegative integer,then the dimension of the family of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions of curves of “complementary” degree s − k.
A last XIXth century version
Theorem (Cayley-Bacharach theorem, version 6) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn respectively, meeting transversely,
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 18 / 1 Set s = ∑di − n − 1. If k ≤ s is a nonnegative integer,then the dimension of the family of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions of curves of “complementary” degree s − k.
A last XIXth century version
Theorem (Cayley-Bacharach theorem, version 6) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn respectively, meeting transversely,and suppose that the 0 intersection Γ = X1 ∩ ... ∩ Xn is the disjoint union of subsets Γ and Γ00.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 18 / 1 then the dimension of the family of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions of curves of “complementary” degree s − k.
A last XIXth century version
Theorem (Cayley-Bacharach theorem, version 6) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn respectively, meeting transversely,and suppose that the 0 intersection Γ = X1 ∩ ... ∩ Xn is the disjoint union of subsets Γ 00 and Γ . Set s = ∑di − n − 1. If k ≤ s is a nonnegative integer,
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 18 / 1 is equal to the failure of Γ00 to impose independent conditions of curves of “complementary” degree s − k.
A last XIXth century version
Theorem (Cayley-Bacharach theorem, version 6) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn respectively, meeting transversely,and suppose that the 0 intersection Γ = X1 ∩ ... ∩ Xn is the disjoint union of subsets Γ 00 and Γ . Set s = ∑di − n − 1. If k ≤ s is a nonnegative integer,then the dimension of the family of curves of degree k containing Γ0 (modulo those containing all of Γ)
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 18 / 1 A last XIXth century version
Theorem (Cayley-Bacharach theorem, version 6) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn respectively, meeting transversely,and suppose that the 0 intersection Γ = X1 ∩ ... ∩ Xn is the disjoint union of subsets Γ 00 and Γ . Set s = ∑di − n − 1. If k ≤ s is a nonnegative integer,then the dimension of the family of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to the failure of Γ00 to impose independent conditions of curves of “complementary” degree s − k.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 18 / 1 The XXth century
Definition Let A be an Artinian ring with residue field C. The ring A is Gorenstein if there exists a C-linear map A → C such that the composition Q : A × A → A → C where the first map is multiplication in A, is a non degenerate pairing on the C-vector space A.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 19 / 1 Let Γ0 and Γ00 be subschemes residual to one another in Γ,and set s = ∑di − n − 1.If k ≤ s is a nonnegative integer, then the dimension of the familiy of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to he failure of Γ00 to impose independent conditions of curves of complementary degree s − k.
A last version
Theorem (Cayley-Bacharach theorem, version 7) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn and suppose that the intersection Γ = X1 ∩ ... ∩ Xn is zero-dimensional.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 20 / 1 and set s = ∑di − n − 1.If k ≤ s is a nonnegative integer, then the dimension of the familiy of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to he failure of Γ00 to impose independent conditions of curves of complementary degree s − k.
A last version
Theorem (Cayley-Bacharach theorem, version 7) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn and suppose that the intersection Γ = X1 ∩ ... ∩ Xn is zero-dimensional.Let Γ0 and Γ00 be subschemes residual to one another in Γ,
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 20 / 1 If k ≤ s is a nonnegative integer, then the dimension of the familiy of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to he failure of Γ00 to impose independent conditions of curves of complementary degree s − k.
A last version
Theorem (Cayley-Bacharach theorem, version 7) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn and suppose that the intersection Γ = X1 ∩ ... ∩ Xn is zero-dimensional.Let Γ0 and Γ00 be subschemes residual to one another in Γ,and set s = ∑di − n − 1.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 20 / 1 A last version
Theorem (Cayley-Bacharach theorem, version 7) n Let X1,...,Xn be hypersurfaces in P (C) of degrees d1,...,dn and suppose that the intersection Γ = X1 ∩ ... ∩ Xn is zero-dimensional.Let Γ0 and Γ00 be subschemes residual to one another in Γ,and set s = ∑di − n − 1.If k ≤ s is a nonnegative integer, then the dimension of the familiy of curves of degree k containing Γ0 (modulo those containing all of Γ) is equal to he failure of Γ00 to impose independent conditions of curves of complementary degree s − k.
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 20 / 1 Thanks for your attention!
Remi´ Bignalet An introduction to the Cayley-Bacharach theorems 21 / 1