Gromov-Witten invariants of blow-ups

1998/04/08 by Andreas Gathmann, Gathmann, Andreas · 4 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/9804043

openalex publication_date 1998/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of P^r and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on P^r having specified global multiplicities or tangent directions in the blown-up points. We give various numerical examples, including a new easy way to determine the famous multiplicity d^-3 for d-fold coverings of rational curves on the quintic threefold, and, as an outlook, two examples of blow-ups along subvarieties, whose Gromov-Witten invariants lead to classical multisecant formulas.

Cited by

• The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants
• Real Gromov-Witten invariants on the moduli space of genus 0 stable maps\n to a smooth rational projective space
• The number of plane conics 5-fold tangent to a given curve
• Degenerations of Del Pezzo surfaces and the Gromov-Witten invariants of the Hilbert Scheme of conics

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