Enumerative Geometry Steven Finch. February 24, 2014

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1 Enumerative Geometry Steven Finch February 24, 204 Given a complex projective variety V (as defined in []), we wish to count the curves in V that satisfy certain prescribed conditions. Let C f denote complex projective -dimensional space. In our first example, V = C f2, the complex projective plane; in the second and third, V is a general hypersurface in C f of degree 2 3. Call such V a cubic twofold when =3and a quintic threefold when =4. Our interest is in rational curves, which include all lines (degree ), conics (degree 2) and singular cubics (degree 3). No elliptic curves are rational. The word rational here refers to the affine parametrization of the curve a ratio of polynomials andthecurveisofdegree if the polynomials are of degree at most. For instance, the circle =is represented as = 2 + = The lemniscate of Bernoulli has degree 4 andisrepresentedas 4 = = 2 ( 2 ) It is also defined implicitly: = 2 2 The semi- and clearly possesses a singularity (vanishing gradient) at the origin. cubical parabola 2 = 3 and four-petal rose =4 2 2 possess likewise. All rational curves, smooth or not, have genus 0. 0 Copyright c 204 by Steven R. Finch. All rights reserved.

2 Enumerative Geometry Rational Plane Curves Passing Through Points. In the following, we use homogeneous coordinates. Given two distinct points ( ), ( ) in fc 2, there is exactly one line passing through bothbecausethesimultaneoussystem of equations + + =0 { 2} has a unique solution ( ) in C f2 (up to a common scalar). It is a little harder to prove the corresponding result for conics. Given five points ( ) in general position, there is exactly one conic passing through all five via study of =0 { } in C f5. Hence we have = 2 =,where is defined as the number of rational curves in C f2 of degree passing through 3 general points. The quantity 3 turns out to be the critical threshold for our question: less would give an answer of infinity, more would give an answer of zero [2]. Proving that 3 =2involves a heavy dose of algebraic geometry [3, 4]. Credit for this accomplishment (in the mid-800s) is assigned variously to Chasles [5] and Steiner [6]. Kontsevich s famous recursion [7, 8, 9]: = X µ µ = 2 was not found until recently (in 994). Its astonishing proof drew upon ideas not from geometry but from mathematical physics, specifically, quantum field theory and string theory. Other relevant recursions for curve counting are known [7, 0,, 2] but these are too complicated for us to discuss here. The asymptotics for are [, 3] µ (3 )! ( ) as, obtained using a device due to Zagier called the asymp trick. closed-form expression for these constants is known. No 0.2. Lines On a Hypersurface. The fact that exactly 27 lines lie on a cubic twofold in f C 3 is a well-known theorem [4, 5] due to Cayley & Salmon (in 849). Somewhat later, Schubert proved (in 886) that exactly 2875 lines lie on a quintic threefold in f C 4.Thuswehave 3 =27and 4 =2875,where is defined as the

3 Enumerative Geometry 3 number of lines on a general hypersurface in C f of degree 2 3. Expanding on these results, van der Waerden proved (in 933) that à 2 3 Y = ( ) (2 3 + )! ( )! and Zagier [9, 3] obtained asymptotics r µ 27 (2 3) as. In this case, closed-form expressions are available Rational Curves On a Quintic Threefold. The number of conics on a cubic twofold is infinity. Incontrast,thenumberofconicsonaquinticthreefold is Our discussion at this point becomes highly speculative it is merely conjectured (by Clemens [8]) that the number of degree rational curves on a quintic threefold is finite but the following calculations are known to be valid at least for 9. Define 0 (), (), 2 () via power series expansion of a certain hypergeometric function [4]: Q 5 = (5 + ) Q = (5 + = 0()+ () + 2 () 2 + )5 It follows that 0 () = (5)! (!) 5 () = à (5)! (!) 5 5X =+! (a similar expression for 2 () would be good to see). We then define rational numbers recursively from yielding 2 () = () () + 5 ½ { } = = µ 0 ()exp () 0 () ¾

4 Enumerative Geometry 4 Such numbers are examples of Gromov-Witten invariants, which count not only the rational curves we desire, but also capture (unwanted) additional structure [8]. The final step is another recursion [4, 6]: = X 3 yielding { } = = { } It is, again, merely conjectured (by Gopakumar & Vafa [8, 9]) that all numbers obtained in this manner are indeed integers. Much work lies ahead to rigorously confirm everything written here. The asymptotics for remain open Addendum. Let be a cubic twofold and let be the number of rational curves on of degree passing through general points on. Traves [9, 7] gave the values { } = = { } and conjectured that is always finite. A recursive formula for (à la Kontsevich for?) remains open. References [] S. R. Finch, Elliptic curves over Q, unpublished note (2005). [2] I. Strachan, How to count curves: from nineteenth-century problems to twentyfirst-century solutions, RoyalSoc.Lond.Philos.Trans.Ser.A36 (2003) ; MR (2005c:53). [3] L. Caporaso, Counting curves on surfaces: a guide to new techniques and results, European Congress of Mathematics, v.i,proc.996budapestconf.,ed.a.balog, G. O. H. Katona, A. Recski and D. Szász, Birkhäuser, 998, pp ; arxiv:alg-geom/96029; MR (99j:4050). [4] A. Gathmann, Enumerative geometry, [5] E. Arbarello, M. Cornalba and P. A. Griffiths, Geometry of Algebraic Curves, v. II, Springer-Verlag, 20, pp ; MR (202e:4059). [6] S. Lanzat and M. Polyak, Counting real curves with passage/tangency conditions, J. London Math. Soc. 85 (202) ; arxiv:0.686; MR

5 Enumerative Geometry 5 [7] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 64 (994) ; arxiv:hepth/940247; MR29244 (95i:4049). [8] S. Katz, Enumerative Geometry and String Theory, Amer. Math. Soc., 2006, pp. 0, 27 3, 25 43, 92 96; MR (2007i:4054). [9] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A03587, A027363, A06004, A07692, and A [0] C. Itzykson, Counting rational curves on rational surfaces, Internat.J.Modern Phys. B 8 (994) ; MR30568 (96e:406). [] P. Di Francesco and C. Itzykson, Quantum intersection rings, The Moduli Space of Curves, Proc. 994 Texel Island conf., ed. R. Dijkgraaf, C. Faber and G. van der Geer, Birkhäuser Boston, 995, pp. 8 48; arxiv:hep-th/94275v; MR (96k:404a). [2] P. Di Francesco, Enumerative geometry from string theory, Math. Comput. Modelling 26 (997) 75 96; MR (98j:4072). [3] D. B. Grünberg and P. Moree, Sequences of enumerative geometry: congruences and asymptotics, Experim. Math. 7 (2008) ; arxiv:math/060286; MR (200k:49). [4] J. J. O Connor and E. F. Robertson, Cubic surfaces, [5] B. Hunt, TheGeometryofSomeSpecialArithmeticQuotients, Springer-Verlag, 996, pp , ; MR (98c:4033). [6] M. Kontsevich, Enumeration of rational curves via torus actions. The Moduli Space of Curves, Proc. 994 Texel Island conf., ed. R. Dijkgraaf, C. Faber and G. van der Geer, Birkhäuser Boston, 995, pp ; arxiv:hep-th/ ; MR (97d:4077). [7] W. Traves, Conics on the cubic surface,