The genus zero Gromov-Witten invariants of [Sym² P²]

2007/02/08 by Jonathan Wise, Wise, Jonathan · 1 citation
Mathematics · #14N10 #14N35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #math.AG #msc:14N10 #msc:14N35

paper · pdf · doi:10.48550/arxiv.math/0702219

33 pages; mostly rewritten, many errors corrected; all comments welcome

openalex publication_date 2007/02/08 · arxiv created 2008/07/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym² P²], the stack symmetric square of P². This compactifies the moduli space of stable maps from hyperelliptic curves to P², and we show that all genus zero Gromov--Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov--Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d + 1 generic points in P². Comparing our method to that of Graber for calculating the same numbers, we verify an example of the crepant resolution conjecture.

Citations

• Orbifold Cohomology of ADE-singularities
• Enumerative geometry of hyperelliptic plane curves
• Quantum Cohomology and Crepant Resolutions: A Conjecture
• Gromov-Witten theory of Deligne-Mumford stacks

Cited by

• A hyperelliptic Hodge integral

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