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 $p_1,p_2,\dots, p_r$ are on a curve of degree $d\leq 3$. Then we assume that the points are very general and show that $\varepsilon(X_r,L,x)\geq 1$ for any ample line bundle $L$ and any $x\in X_r$ if the Strong SHGH conjecture is true. We explore the relation between bounded negativity and Seshadri constants and study the Seshadri function on $X_r$.