13/2 ways of counting curves

TL;DR

This survey explores 13/2 principal methods for counting curves in algebraic varieties, especially 3-folds, highlighting their interconnections and providing an accessible guide for newcomers.

Contribution

It offers a comprehensive overview of various curve counting theories, emphasizing their relationships and aiming to serve as an introductory resource.

Findings

Multiple curve counting theories are interconnected through conjectural relationships.

The 3-fold case reveals the richest geometric structures.

A unified framework helps in understanding diverse enumerative approaches.

Abstract

In the past 20 years, compactifications of the families of curves in algebraic varieties X have been studied via stable maps, Hilbert schemes, stable pairs, unramified maps, and stable quotients. Each path leads to a different enumeration of curves. A common thread is the use of a 2-term deformation/obstruction theory to define a virtual fundamental class. The richest geometry occurs when X is a nonsingular projective variety of dimension 3. We survey here the 13/2 principal ways to count curves with special attention to the 3-fold case. The different theories are linked by a web of conjectural relationships which we highlight. Our goal is to provide a guide for graduate students looking for an elementary route into the subject.