2002/01/31 by Andreas Gathmann, Gathmann, Andreas
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.math/0202002
16 pages
arxiv created 2002/01/31 · openalex publication_date 2002/01/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a general plane curve Y of degree d, we compute the number nd of irreducible plane conics that are 5-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved there because the calculations received too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number nd in terms of relative Gromov-Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of P2 branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in P2, and compare this number to the corresponding Yau-Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.