Lower bounds for Seshadri constants on blow ups of $\mathbb{P}^2$

Abstract

Let $\pi: X_r \rightarrow \mathbb P^2$ be a blow up of $\mathbb P^2$ 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 $\varepsilon(X_r)\geq \frac{1}{2}$ if the Strong SHGH conjecture is true.