The Caporaso-Harris-Ran degeneration principle: proof and applications

TL;DR

This paper provides a complete proof of the Caporaso-Harris recursive formula for counting algebraic curves, along with applications to degenerations of Severi varieties and enumerative geometry.

Contribution

It offers a rigorous proof of the Caporaso-Harris degeneration principle and explores its applications in enumerating curves on degenerating surfaces.

Findings

Proof of the Caporaso-Harris recursive formula

Description of limits of Severi varieties under surface degenerations

Applications to curve enumeration in algebraic geometry

Abstract

Severi varieties are the parameter spaces for curves with prescribed homology class and genus on a smooth surface. We describe their limits along degenerations of surfaces, with a view towards the enumeration of curves. This includes a complete proof of the Caporaso-Harris recursive formula, with all the necessary background on deformations of curves and singularities.