Fixed-domain curve counts for blow-ups of projective space

2023/03/06 by Cela, Alessio, Lian, Carl · 4 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.03433

Abstract

We study the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual (Gromov-Witten) counts are found to agree asymptotically in the Fano (and some (-K)-nef) examples, but not in general. For toric blow-ups, geometric counts are expressed in terms of integrals on products of Jacobians and symmetric products of the domain curves, and evaluated explicitly in genus 0 and in the case of Bl_q(ℙ^r). Virtual counts for Bl_q(ℙ^r) are also computed via the quantum cohomology ring.

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• Intersection Theory of Hyperquot Schemes on curves
• Curves in projective space and RSK
• Enumerating log rational curves on some toric varieties
• A short way of counting maps to hypersurfaces in Grassmannians

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