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

Alessio Cela, Carl Lian

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 $\text{Bl}_q(\mathbb{P}^r)$. Virtual counts for $\text{Bl}_q(\mathbb{P}^r)$ are also computed via the quantum cohomology ring.

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

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 -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 . Virtual counts for are also computed via the quantum cohomology ring.
Paper Structure (30 sections, 46 theorems, 169 equations, 2 figures)

This paper contains 30 sections, 46 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Curve-counting with fixed domain
  3. Curve counts on $\mathbb{P}^r$
  4. New results
  5. Enumerativity
  6. Geometric calculations
  7. Virtual calculations
  8. Acknowledgments
  9. Enumerativity
  10. Failure of SAE
  11. Generalities for Fano varieties
  12. Blow-ups of $\mathbb{P}^r$
  13. SAE for ${\mathsf{Bl}}_q(\mathbb{P}^r)$
  14. SAE for del Pezzo surfaces
  15. SAE in dimension $3$
  16. ...and 15 more sections

Key Result

Lemma 1.1

Let $Z\subset {\mathcal{M}}_{g,n}(X,\beta)$ be an irreducible component. Suppose that $Z$ dominates ${\mathcal{M}}_{g,n}\times X^n$ and that $n\ge g+1$. Then, $Z$ is generically smooth of dimension equal to the virtual dimension. More generally, let $Z\subset \overline{\mathcal{M}}_{g,n}(X,\beta)$ b

Figures (2)

  • Figure 1: Topological type of the domain curves of points in ${\mathcal{M}}_\Gamma^{(m,a)}$
  • Figure 2: Topological type of the domain curves of points in ${\mathcal{M}}_\Gamma^{(m,s,a)}$

Theorems & Definitions (93)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Proposition 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 1.10: Integral formula
  • Theorem 1.11
  • ...and 83 more