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

TL;DR

This paper computes Seshadri constants on b^1b^1 and applies these results to explicitly solve the symplectic packing problem, revealing differences based on the parity of the number of points considered.

Contribution

It provides explicit formulas for Seshadri constants on b^1b^1 and introduces a new reflection method for analyzing these constants outside Mori's cone.

Findings

Explicit formulas for symplectic packing on b^1b^1.

A new reflection method for computing Seshadri constants.

Distinct behaviors for odd and even numbers of points.

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.