Enumerative geometry of plane curves of low genus

Details Enumerative geometry of plane curves of low genus Enumerative geometry of plane curves of low genus

1998-03-02

arxiv.org/abs/math/9803007v1

Mathematics

Algebraic Geometry

23 pages, LaTeX

Scientific paper

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande.

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