A simple proof that rational curves on K3 are nodal

By Xi Chen. 2000

73 citations recorded by the source corpus.

The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fibration over P with a unique section C, fibers F and λ = 2. Here λ = c1(π∗ω) is the first Chern class of the Hodge bundle π∗ω of the fibration π : S → P (see [H-M]). It is a standard result that the number of nodal fibers of an elliptic fibration are given by 12λ [H-M, p. 158]. So there are exactly 24 rational nodal curves in the linear series |F | for S general. This is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We will call a K3 surface with Picard lattice (1.1) a BL K3 surface. A BL K3 surface S lies on the boundary of the moduli space of K3 surfaces of genus g with C+gF as the corresponding primitive divisor. Every curve in the linear series |OS(C + gF )| is “totally reducible”, i.e., it consists of the −2 curve C and g elliptic “tails” attached to C. A curve D ∈ |OS(C + gF )| is the image of a stable rational map only if D = C ∪m1F1∪m2F2∪ ...∪m24F24, where F1, F2, ..., F24 are 24

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/df57c69d4c592eff5e35c04127bf263dada871fc

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Citations in the corpus

Lectures on K3 Surfaces (2016, background): “which are contained in |nH| with n > 1? These questions have been dealt with in Chen’s papers [112] and [113].”
Curves and cycles on K3 surfaces (2013, background): “The key point is eventually the algebraicity of the zero-locus of normal functions recently established by Brosnan–Pearlstein [BP09] and M. Saito [Sa08], see [Cha13] for a recent overview.”
A primer on Seshadri constants (2008, result): “e rational curve C2jLj(resp. C2j2Lj) such that Chas an ordinary singular point of multiplicity +1 (resp. 2+1) at x and is smooth outside x, and "(L) = LC=mult xC. By a well-known result of Chen [Che02], rational curves in the primitive class of a general K3 surface in the moduli space are nodal”
Refined curve counting on complex surfaces (2012, background): “Remark 48 The assumption on genericity of nodal curves in Theorem 47 holds for a locally versal family by [17; 69], for the general P ı jLj when L is ı–very-ample by [43], for the general P ı jLj when L is irreducible on a general K3 surface (the genericity of nodal curves in maximal cogenus is show”