Enumerative geometry of plane curves of low genus

Abstract

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.

arXiv comment23 pages, LaTeX

References in corpus (6)

• Topological recursion relations in genus 2
• Large $N$ Phases of Chiral QCD_2
• Localization of virtual classes
• Recursive Formulas for the Characteristic Numbers of Rational Plane Curves
• A geometric construction of Getzler's relation
• Composition law and Nodal genus-2 curves in P^2

Cited by in corpus (6)

• Hurwitz numbers and intersections on moduli spaces of curves
• Enumeration of Genus-Two Curves with a Fixed Complex Structure in P^2 and P^3
• Linear sections of the Severi variety and moduli of curves
• Enumeration of plane unicuspidal curves of any genus via tropical geometry
• Characteristic numbers of rational curves with cusp or prescribed triple contact
• Descendant invariants and characteristic numbers