Counting rational curves on K3 surfaces

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Counting rational curves on K3 surfaces Arnaud BEAUVILLE 1 Introduction The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1 ; a surface in Fg admits a g- dimensional linear system of curves of genus g . A naıve count of constants suggests that such a system will contain a positive number, say n(g) , of rational (highly singular) curves. The formula is ∑ g≥0 n(g)qg = q ∆(q) , where ∆(q) = q ∏ n≥1(1? q n)24 is the well-known modular form of weight 12 , and we put by convention n(0) = 1 . To explain the idea in a nutshell, take the case g = 1 . We are thus looking at K3 surfaces with an elliptic fibration f : S ? P1 , and we are asking for the number of singular fibres. The (topological) Euler-Poincare characteristic of a fibre Ct is 0 if Ct is smooth, 1 if it is a rational curve with one node, 2 if it has a cusp, etc. From the standard properties of the Euler-Poincare characteristic, we get e(S) = ∑ t e(Ct) ; hence n(1) = e(S) = 24 ,

• jacobian j¯ct


• local invariant


• any unibranch partial


• fibre


• bijective homomorphism


• homomorphism o?


• o?c ?


• rational curve


• o? c˜

Sujets

• Europe
• Beauville
• Taylor
• Algebraic curve

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 16
• Langue: English

Extrait

Counting rational curves on K3 surfaces

1 Arnaud B EAUVILLE

Introduction The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (F g ) g≥1 ; a surface inF g admits ag- dimensional linear system of curves of genusgggsutsantsesı¨evocnuotcfnots.Ana that such a system will contain a positive number, sayn(g) , of rational (highly singular) curves. The formula is X q g n(g)q=, Δ(q) g≥0 Q n24 where Δ(q) =q(1−q) is the well-known modular form of weight 12 , and n≥1 we put by conventionn(0) = 1 . To explain the idea in a nutshell, take the casegWe are thus looking= 1 . 1 at K3 surfaces with an elliptic fibrationf: S→P, and we are asking for the numberofsingularfibres.The(topological)Euler-Poincar´echaracteristicofafibre C t is 0 if C t it is a rational curve with one node, 2 if it has ais smooth, 1 if cusp,etc.FromthestandardpropertiesoftheEuler-Poincare´characteristic,weget X e(S) =e(C t hence) ; n(1) =e(S) = 24 , and this number counts nodal rational t curves with multiplicity 1 , cuspidal rational curves with multiplicity 2 , etc. The idea of Yau and Zaslow is to generalize this approach to any genus. Let S be a K3 surface with ag-dimensional linear system Π of curves of genusg. The ¯ role offwill be played by the morphism→J C Π whose fibre over a pointt∈Π ¯ is the compactified Jacobian JC t apply the same method, we would like to. To prove the following facts: ¯ g 1)TheEuler-Poincar´echaracteristice(J C) is the coefficient ofqin the Taylor expansion ofq/Δ(q) . ¯ 2)e(JC t if C) = 0 t is not rational. ¯ 3)e(JC t ) = 1 if C t Moreoveris a rational curve with nodes as only singularities. ¯ e(JC t Cpositive when ) is t is rational, and can be computed in terms of the singularities of C t . 4) For a generic K3 surface S inF g , all rational curves in Π are nodal. ¯ The first statement is proved in§1, by comparinge(J C) with the Euler- [g] Poincar´echaracteristicoftheHilbertschemeSwhichhasbeencomputedby

1 Partially supported by the European HCM project “Algebraic Geometry in Europe” (AGE).

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