Lower bounds for Seshadri constants on blow ups of P2

Abstract

Let $\pi :X_r\to P^2$ be a blow up of $P^2$ at r distinct points $p_1,p_2,\dots ,p_r$. We study lower bounds for Seshadri constants of ample line bundles on $X_r$. First, we consider the case when the points lie on a curve of degree $d\le 3$, and the case when $r\le 8$. We then assume that the points are very general and show that $\epsilon \left(X_r\right)\ge \frac{1}{2}$ if the Strong SHGH conjecture is true.