Rationality of Seshadri constants on general blow ups of P²

Let X be a projective surface and let L be an ample line bundle on X. The global Seshadri constant ε(L) of L is defined as the infimum of Seshadri constants ε(L,x) as x∈ X varies. It is an interesting question to ask if ε(L) is a rational number for any pair $(X, L)$. We study this question when X is a blow up of P² at r ≥ 0 very general points and L is an ample line bundle on X. For each r we define a submaximality threshold which governs the rationality or irrationality of ε(L). We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.