13/2 ways of counting curves

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.

Typo fixed, In "Moduli spaces", LMS Lecture Note Series, 411 (2014), 282-333. Cambridge University Press

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Cited by in corpus (20)

• Donaldson-Thomas theory for Calabi-Yau 4-folds
• Curve counting on elliptic Calabi-Yau threefolds via derived categories
• Gopakumar-Vafa invariants do not determine flops
• Virtual classes for the working mathematician
• Notes on the proof of the KKV conjecture
• Non-archimedean quantum K-invariants
• Rank-one sheaves and stable pairs on surfaces
• Reduced Donaldson-Thomas invariants and the ring of dual numbers
• Homology of Hilbert schemes of points on a locally planar curve
• Virasoro constraints for stable pairs on toric 3-folds
• Hall algebras and Donaldson-Thomas invariants
• EGL formula for DT/PT theory of local curves
• Noncommutative resolutions and CICY quotients from a non-abelian GLSM
• Multiple Cover Calculation for the Unramified Compactification of the Moduli Space of Stable Maps
• On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve
• Virtual Poincaré polynomial of the space of stable pairs supported on quintic curves
• $P=W$ phenomena in algebraic and enumerative geometry
• Double-dimer condensation and the PT-DT correspondence
• Symmetric periodic orbits and uniruled real Liouville domains
• Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I