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Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem

Chris Dionne, Mike Roth

TL;DR

The paper computes r-point Seshadri constants ε_r(L) for line bundles on Y = P^1×P^1, distinguishing inner and outer bundles via the roots α_r,β_r of t^2−((r−4)/2)t+1=0. A central tool is the square-zero cone in the V_r subspace, with an infinite-order automorphism T_r for even r that generates a family of nef square-zero boundary classes, enabling explicit ε_r(L) formulas (outer bundles) and, for odd r, a finite boundary governed by four (or five) (−1)-curve types. The authors develop two principal methods to produce inner square-zero nef classes: Petrakiev reflections (section 8–9) and pullbacks via degree maps φ_{a,b} (section 10), yielding new infinite families of inner nef classes and sharp obstructions for symplectic packing via ν_r(L). They apply these geometric/diophantine techniques to derive complete packing constants ν_r(L) for all real ample L and r≥9 (with small-r cases treated separately), providing exact criteria for full packings and revealing a striking even/odd dichotomy. Overall, the work combines square-zero geometry, automorphisms, and reflection/pullback constructions to produce explicit Seshadri constants and symplectic packing results on the blowups of P^1×P^1, with potential implications for irrational Seshadri constants and nef-cone structure.

Abstract

In this paper we compute the $r$-point Seshadri constant on $\mathbb{P}^1\times\mathbb{P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb{P}^1\times\mathbb{P}^1$. Some exact values of the Seshadri constant outside the region governed by Mori's cone theorem are also given. These latter results use a useful new "reflection method". In the analysis there is a striking difference between the cases when $r$ is odd and when $r$ is even. When $r$ is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$-curves to consider. In contrast, when $r$ is odd only a finite number (usually $4$) types of $(-1)$-curves are relevant to our answer.

Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, and applications to the symplectic packing problem

TL;DR

The paper computes r-point Seshadri constants ε_r(L) for line bundles on Y = P^1×P^1, distinguishing inner and outer bundles via the roots α_r,β_r of t^2−((r−4)/2)t+1=0. A central tool is the square-zero cone in the V_r subspace, with an infinite-order automorphism T_r for even r that generates a family of nef square-zero boundary classes, enabling explicit ε_r(L) formulas (outer bundles) and, for odd r, a finite boundary governed by four (or five) (−1)-curve types. The authors develop two principal methods to produce inner square-zero nef classes: Petrakiev reflections (section 8–9) and pullbacks via degree maps φ_{a,b} (section 10), yielding new infinite families of inner nef classes and sharp obstructions for symplectic packing via ν_r(L). They apply these geometric/diophantine techniques to derive complete packing constants ν_r(L) for all real ample L and r≥9 (with small-r cases treated separately), providing exact criteria for full packings and revealing a striking even/odd dichotomy. Overall, the work combines square-zero geometry, automorphisms, and reflection/pullback constructions to produce explicit Seshadri constants and symplectic packing results on the blowups of P^1×P^1, with potential implications for irrational Seshadri constants and nef-cone structure.

Abstract

In this paper we compute the -point Seshadri constant on for those line bundles where the answer might be expected to be governed by -curves. As a consequence we obtain explicit formulas for the symplectic packing problem for . Some exact values of the Seshadri constant outside the region governed by Mori's cone theorem are also given. These latter results use a useful new "reflection method". In the analysis there is a striking difference between the cases when is odd and when is even. When is even the problem admits an infinite order automorphism, and there are infinitely many -curves to consider. In contrast, when is odd only a finite number (usually ) types of -curves are relevant to our answer.
Paper Structure (10 sections, 81 equations)

This paper contains 10 sections, 81 equations.

Table of Contents

  1. Introduction
  2. The square-zero cone and graphical arguments
  3. Even $r$, $r\geqslant 10$
  4. Odd $r$, $r\geqslant 9$
  5. Small $r$ ($1\leqslant r\leqslant 8$)
  6. A brief study of the slopes of the $\xi_n$
  7. Other arguments related to the symplectic packing problem
  8. Petrakiev reflections
  9. Inner square-zero nef classes via reflections
  10. Inner square-zero nef classes via pullbacks