Seshadri constants on P¹× P¹ and applications to the symplectic packing problem

In this paper we compute the r -point Seshadri constant on P¹× P¹ 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 P¹× P¹ . Some exact values for the r -point 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 four) types of $(-1)$ -curves are relevant to our answer.