2011/11/30 by Rahul Pandharipande, R. Pandharipande, Richard Thomas +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Computer science #Content (measure theory) #Mathematical analysis #Mathematics #Mathematics and Applications #Polynomial and algebraic computation #hep-th #math.AG #math.SG #msc:14N #msc:14N35
paper · pdf · doi:10.1017/cbo9781107279544.007
Typo fixed, In "Moduli spaces", LMS Lecture Note Series, 411 (2014), 282-333. Cambridge University Press
openalex publication_date 2014/03/13 · arxiv created 2016/05/09 · arxiv updated 2016/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.