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Upper bounds for logarithmic Gromov-Witten invariants of projective space

Dan Simms

Abstract

We provide an upper bound for the genus zero logarithmic Gromov-Witten invariants of projective space relative to its toric boundary. The upper bound is polynomial in the contact orders, with degree depending on the number of marked points. The result hinges on the positivity of intersections for projective spaces.

Upper bounds for logarithmic Gromov-Witten invariants of projective space

Abstract

We provide an upper bound for the genus zero logarithmic Gromov-Witten invariants of projective space relative to its toric boundary. The upper bound is polynomial in the contact orders, with degree depending on the number of marked points. The result hinges on the positivity of intersections for projective spaces.
Paper Structure (13 sections, 9 theorems, 47 equations, 3 figures, 4 tables)

This paper contains 13 sections, 9 theorems, 47 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main Result
  3. Proof Outline
  4. Future Work
  5. Acknowledgments
  6. Background
  7. Logarithmic Gromov--Witten Theory of Toric Varieties
  8. Positivity
  9. Proof of Theorem \ref{['Main']}
  10. Transversality
  11. Alternative Compactification
  12. Calculations
  13. Examples

Key Result

Theorem 1.1

For each marking $i$, let $\upalpha^{\max}_i = \max_j \upalpha_{ij}.$ Then where $\upnu$ is the maximal codimension of an insertion at a marked point with no tangency to the boundary. In particular, if one imposes a point constraint at such a point this binomial coefficient is 1.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (27)

  • Theorem 1.1
  • Example 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • ...and 17 more