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Cremona symmetry in Gromov-Witten theory

Amin Gholampour, Dagan Karp, Sam Payne

TL;DR

The paper introduces a Cremona-based symmetry in the genus-0 Gromov-Witten theory of ${\bf P}^n$ and its point-blowups, realized via a toric symmetry on the permutohedral variety $X_{\Pi_n}$. By resolving the Cremona transform through an iterated toric blowup and exploiting functoriality of GW invariants, the authors derive an explicit transformation on curve data $(d,\{a_i\})$ that preserves stationary invariants for nonexceptional classes, enabling computation of otherwise hard invariants from simpler ones. This framework yields a practical computational tool and provides a GW-theoretic proof of the classical uniqueness of the rational normal curve, via degeneration from the permutohedral setting back to ${\bf P}^n$ and its blowups. The approach hinges on toric geometry of the permutohedron, degeneration to the normal cone, and a careful analysis of relative GW theory to identify nonexceptional classes, thereby linking intricate invariants to classical enumerative geometry results with potential broader applicability.

Abstract

We establish the existence of a symmetry within the Gromov-Witten theory of $\mathbb{CP}^n$ and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants.

Cremona symmetry in Gromov-Witten theory

TL;DR

The paper introduces a Cremona-based symmetry in the genus-0 Gromov-Witten theory of and its point-blowups, realized via a toric symmetry on the permutohedral variety . By resolving the Cremona transform through an iterated toric blowup and exploiting functoriality of GW invariants, the authors derive an explicit transformation on curve data that preserves stationary invariants for nonexceptional classes, enabling computation of otherwise hard invariants from simpler ones. This framework yields a practical computational tool and provides a GW-theoretic proof of the classical uniqueness of the rational normal curve, via degeneration from the permutohedral setting back to and its blowups. The approach hinges on toric geometry of the permutohedron, degeneration to the normal cone, and a careful analysis of relative GW theory to identify nonexceptional classes, thereby linking intricate invariants to classical enumerative geometry results with potential broader applicability.

Abstract

We establish the existence of a symmetry within the Gromov-Witten theory of and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants.

Paper Structure

This paper contains 18 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Summary
  4. Preliminaries
  5. Main Results
  6. The Cremona Transform
  7. Toric Construction
  8. The base space $\mathbb{P}^{n}$
  9. The blowup at points, $X_{0}$
  10. The blowup at lines, $X_{1}$
  11. The full permutohedral case, $\widehat{X} (n+1)$
  12. The general case, $\widehat{X} (m)$
  13. Toric Symmetry
  14. Proof of Main Theorem
  15. Nonexceptional classes
  16. ...and 3 more sections

Key Result

Theorem 1

Let $X$ be the blowup of $\mathbb{P}^{n}$ along $m$ general points $( p_{1},\dotsc ,p_{m}) \in \mathbb{P}^{n}$, where $m>n+1$. Let $h \in H_{2} (X; {\mathbb Z})$ denote the pullback of the class of a general line in $\mathbb{P}^{n}$, and let $e_{i} \in H_{2} (X)$ denote the class of a line in the ex where $\beta '= d'h-\sum a_{i}' e_{i}$, and $d', a_{i}' \in {\mathbb Z}$ are given by:

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 2
  • Remark 3
  • Proposition 4: Bryan-Leung
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • ...and 2 more