Reduced classes and curve counting on surfaces I: theory

TL;DR

This paper develops a new theory of reduced Gromov-Witten and stable pair invariants for surfaces, connecting classical enumerative geometry with modern conjectures and enabling advanced calculations.

Contribution

It introduces a novel framework for reduced invariants on surfaces, proving a case of the MNOP conjecture and extending the G"ottsche conjecture to non-ample cases.

Findings

Classical Severi degrees are special cases of the new invariants.

Proves a case of the MNOP conjecture.

Identifies a property of the moduli space of stable pairs enabling calculations.

Abstract

We develop a theory of \emph{reduced} Gromov-Witten and stable pair invariants of surfaces and their canonical bundles. We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP conjecture, and allows us to generalise the G\"ottsche conjecture to the non-ample case. In a sequel we prove this generalisation. We prove a remarkable property of the moduli space of stable pairs on a surface. It is the zero locus of a section of a bundle on a smooth compact ambient space, making calculation with the reduced virtual cycle possible.