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

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.