Refined curve counting on complex surfaces

TL;DR

This paper introduces refined invariants for counting nodal curves on complex surfaces, proposes conjectures on their generating functions and formulas for special surfaces, and refines existing recursion methods linking complex and real enumerative invariants.

Contribution

It defines new refined invariants for curve counting, conjectures their multiplicative generating functions, and refines the Caporaso-Harris recursion to unify complex and real enumerative invariants.

Findings

Refined invariants for nodal curves are defined and conjectured to have multiplicative generating functions.

A refinement of the Caporaso-Harris recursion is proposed, conjectured to produce the same invariants.

Specialization at y = -1 links to Welschinger invariants for real enumerations.

Abstract

We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces. We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at y = -1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.