Rationality of Seshadri constants on general blow ups of ${\mathbb{P}}^2$

Abstract

Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\epsilon (L)$ of $L$ is defined as the infimum of Seshadri constants $\epsilon (L,x)$ as $x\in X$ varies. It is an interesting question to ask if $\epsilon (L)$ is a rational number for any pair $(X,L)$. We study this question when $X$ is a blow up of ${\mathbb{P}}^2$ at $r\ge 0$ very general points and $L$ is an ample line bundle on $X$. For each $r$ we define a $\text{submaximality threshold}$ which governs the rationality or irrationality of $\epsilon (L)$. We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.

Proof of Theorem 4.1 re-organized for more clarity; some other minor changes; to appear in J. Pure Appl. Algebra