LAUSR

Abstract Let X be a projective surface and let L be an ample line bundle on X. The global Seshadri constant e ( L ) of L is defined as the infimum of Seshadri constants e ( L , x ) as x ∈ X varies. It is an interesting question to ask if e ( L ) is a rational number for any pair ( X , L ) . We study this question when X is a blow up of P 2 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 e ( L ) . We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.