Curve Counting \`a la G\"ottsche

TL;DR

This paper surveys the problem of understanding the enumeration of elta-nodal curves on algebraic surfaces, focusing on Gottsche's conjectures and recent refinements, highlighting open problems and progress.

Contribution

It reviews five key subproblems related to Gottsche's conjectures and recent developments in counting nodal curves on algebraic surfaces.

Findings

Overview of Gottsche's conjectures and their status

Identification of five open subproblems in curve counting

Summary of recent progress and research directions

Abstract

Let n_\delta be the number of \delta-nodal curves lying in a suitably ample complete linear system |L| and passing through appropriately many points on a smooth projective complex algebraic surface. A major open problem is to understand the behavior of n_\delta, specifically to finish off Lothar G\"ottsche's mostly proved 1997 conjectures and then go on to treat the new refinements by G\"ottsche and Vivek Shende. Five subproblems are explained, and work on them is surveyed.