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Classical Enumerative Geometry and Quantum Cohomology Classical Enumerative Geometry and Quantum Cohomology James McKernan UCSB Classical Enumerative Geometry and Quantum Cohomology – p.1 Amazing Equation It is the purpose of this talk to convince the listener that the following formula is truly amazing. end a − a − b c 2 2 3 − 3 3 Nd(a; b; c) = Nd1 (a1; b1; c1)Nd2 (a2; b2; c2) d1d2 d1d2 a1 − 1 a1 b1 c1 d1+Xd2=d a1+a2=a−1 b1+b2=b c1+c2=c a − a − b c · 2 3 − 3 3 + 2 Nd1 (a1; b1; c1)Nd2 (a2; b2; c2) d1d2 d1 a1 − 1 a1 b1 b2 1 c1 d1+Xd2=d a1+a2=a b1+b2=b−1 c1+c2=c a − a − b c · 3 − 2 3 + 4 Nd1 (a1; b1; c1)Nd2 (a2; b2; c2) d1d2 d1 a1 − 2 a1 − 1 b1 b2 2 c1 d1+Xd2=d a1+a2=a+1 b1+b2=b−2 c1+c2=c a − a − b c · 3 − 2 3 + 2 Nd1 (a1; b1; c1)Nd2 (a2; b2; c2) d1d2 d1 : a1 − 2 a1 − 1 b1 c1 c2 1 d1+Xd2=d a1+a2=a+1 b1+b2=b Classical Enumerative Geometry and Quantum Cohomology – p.2 c1+c2=c−1 Suppose n = 1. Pick p = (x; y) 2 C2. The line through p is represented by its slope, that is the ratio z = y=x. We get the classical Riemann sphere, C compactified by adding a point at infinity. z 2 C [ f1g Projective Space Pn is the set of lines in Cn+1. Classical Enumerative Geometry and Quantum Cohomology – p.3 We get the classical Riemann sphere, C compactified by adding a point at infinity. z 2 C [ f1g Projective Space Pn is the set of lines in Cn+1. Suppose n = 1. Pick p = (x; y) 2 C2. The line through p is represented by its slope, that is the ratio z = y=x. Classical Enumerative Geometry and Quantum Cohomology – p.3 Projective Space Pn is the set of lines in Cn+1. Suppose n = 1. Pick p = (x; y) 2 C2. The line through p is represented by its slope, that is the ratio z = y=x. We get the classical Riemann sphere, C compactified by adding a point at infinity. z 2 C [ f1g Classical Enumerative Geometry and Quantum Cohomology – p.3 Suppose we are given four points p, q, r and s. Then there is a unique Möbius transformation, sending p −! 0, q −! 1, r −! 1. The image s −! λ 2 C − f0; 1g is called the cross-ratio. link. In fact (r − q)(p − s) λ = : (p − q)(r − s) Automorphisms of P1 P1 az+b Aut( ) = cz+d, the group of Möbius transformations. Classical Enumerative Geometry and Quantum Cohomology – p.4 The image s −! λ 2 C − f0; 1g is called the cross-ratio. link. In fact (r − q)(p − s) λ = : (p − q)(r − s) Automorphisms of P1 P1 az+b Aut( ) = cz+d, the group of Möbius transformations. Suppose we are given four points p, q, r and s. Then there is a unique Möbius transformation, sending p −! 0, q −! 1, r −! 1. Classical Enumerative Geometry and Quantum Cohomology – p.4 link. In fact (r − q)(p − s) λ = : (p − q)(r − s) Automorphisms of P1 P1 az+b Aut( ) = cz+d, the group of Möbius transformations. Suppose we are given four points p, q, r and s. Then there is a unique Möbius transformation, sending p −! 0, q −! 1, r −! 1. The image s −! λ 2 C − f0; 1g is called the cross-ratio. Classical Enumerative Geometry and Quantum Cohomology – p.4 Automorphisms of P1 P1 az+b Aut( ) = cz+d, the group of Möbius transformations. Suppose we are given four points p, q, r and s. Then there is a unique Möbius transformation, sending p −! 0, q −! 1, r −! 1. The image s −! λ 2 C − f0; 1g is called the cross-ratio. link. In fact (r − q)(p − s) λ = : (p − q)(r − s) Classical Enumerative Geometry and Quantum Cohomology – p.4 In fact these two curves intersect at two points and these two lines meet at a point. Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Some Consequences Our definition of Pn has some interesting consequences. Classical Enumerative Geometry and Quantum Cohomology – p.5 intersect at two points and these two lines meet at a point. Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Some Consequences Our definition of Pn has some interesting consequences. In fact these two curves Classical Enumerative Geometry and Quantum Cohomology – p.5 and these two lines meet at a point. Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Some Consequences Our definition of Pn has some interesting consequences. In fact these two curves intersect at two points Classical Enumerative Geometry and Quantum Cohomology – p.5 meet at a point. Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Some Consequences Our definition of Pn has some interesting consequences. In fact these two curves intersect at two points and these two lines Classical Enumerative Geometry and Quantum Cohomology – p.5 Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Some Consequences Our definition of Pn has some interesting consequences. In fact these two curves intersect at two points and these two lines meet at a point. Classical Enumerative Geometry and Quantum Cohomology – p.5 Some Consequences Our definition of Pn has some interesting consequences. In fact these two curves intersect at two points and these two lines meet at a point. Principle of Continuity The number of intersection points is an invariant of a continuous family of curves. Classical Enumerative Geometry and Quantum Cohomology – p.5 Indeed the result is obviously true when our two curves are the union of d and e lines Proof. Let F1 and G1 be the product of linear forms. Then F + tF1 and G + tG1 define continuous families and when t = 1, the answer is obviously de. Bézout’s Theorem Theorem. (Bezout’s Theorem) Given two plane curves C and D defined by polynomials F and G of degree d and e, then #C \ D = de: Classical Enumerative Geometry and Quantum Cohomology – p.6 Proof. Let F1 and G1 be the product of linear forms. Then F + tF1 and G + tG1 define continuous families and when t = 1, the answer is obviously de. Bézout’s Theorem Theorem. (Bezout’s Theorem) Given two plane curves C and D defined by polynomials F and G of degree d and e, then #C \ D = de: Indeed the result is obviously true when our two curves are the union of d and e lines Classical Enumerative Geometry and Quantum Cohomology – p.6 Bézout’s Theorem Theorem. (Bezout’s Theorem) Given two plane curves C and D defined by polynomials F and G of degree d and e, then #C \ D = de: Indeed the result is obviously true when our two curves are the union of d and e lines Proof. Let F1 and G1 be the product of linear forms. Then F + tF1 and G + tG1 define continuous families and when t = 1, the answer is obviously de. Classical Enumerative Geometry and Quantum Cohomology – p.6 Bézout’s Theorem: If we have n hypersurfaces X1; X2; : : : ; Xn of degrees d1; d2; : : : ; dn, then the number of common intersection points is d1d2 : : : dn: Indeed, the same proof applies. The degree The degree d of a variety Xk ⊂ Pn is the number of points X \ Λ, where Λ is a general linear subspace of dimension n − k. Classical Enumerative Geometry and Quantum Cohomology – p.7 Indeed, the same proof applies. The degree The degree d of a variety Xk ⊂ Pn is the number of points X \ Λ, where Λ is a general linear subspace of dimension n − k. Bézout’s Theorem: If we have n hypersurfaces X1; X2; : : : ; Xn of degrees d1; d2; : : : ; dn, then the number of common intersection points is d1d2 : : : dn: Classical Enumerative Geometry and Quantum Cohomology – p.7 The degree The degree d of a variety Xk ⊂ Pn is the number of points X \ Λ, where Λ is a general linear subspace of dimension n − k. Bézout’s Theorem: If we have n hypersurfaces X1; X2; : : : ; Xn of degrees d1; d2; : : : ; dn, then the number of common intersection points is d1d2 : : : dn: Indeed, the same proof applies. Classical Enumerative Geometry and Quantum Cohomology – p.7 How many conics pass through five points, p1, p2, p3, p4 and p5? The condition that a conic contains a point p is a linear condition on the coefficients. Conics in P2 A conic is given as the zero locus of aX2 + bY 2 + cZ2 + dXY + eY Z + fXZ; where [a : b : c : d : e : f] 2 P5. Classical Enumerative Geometry and Quantum Cohomology – p.8 The condition that a conic contains a point p is a linear condition on the coefficients. Conics in P2 A conic is given as the zero locus of aX2 + bY 2 + cZ2 + dXY + eY Z + fXZ; where [a : b : c : d : e : f] 2 P5. How many conics pass through five points, p1, p2, p3, p4 and p5? Classical Enumerative Geometry and Quantum Cohomology – p.8 Conics in P2 A conic is given as the zero locus of aX2 + bY 2 + cZ2 + dXY + eY Z + fXZ; where [a : b : c : d : e : f] 2 P5.
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