An invitation to the enumerative geometry of degenerations

TL;DR

This paper introduces logarithmic Gromov-Witten theory as a tool for studying the enumerative geometry of degenerations, connecting the GW theory of smooth varieties with that of their degenerations via logarithmic methods.

Contribution

It surveys approaches to constructing moduli spaces for stable maps in logarithmic GW theory and explains how these relate to the GW theory of degenerations.

Findings

Logarithmic GW theory effectively relates smooth and degenerate fibers.

Multiple approaches to moduli space construction are discussed.

Applications include insights into tautological classes on moduli spaces.

Abstract

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to constructing well-behaved, virtually smooth moduli spaces of stable maps to such degenerations. Each irreducible component of the special fiber of a degeneration determines a pair consisting of a variety and a normal crossings divisor, and these pairs carry their own logarithmic GW theory. We explain how the GW theory of the general fiber can be expressed in terms of the logarithmic GW theory of these pairs. Finally, we discuss applications to tautological classes on the moduli space of curves.