Lower bounds for Seshadri constants on blow ups of P²

Let π: Xᵣ → P² be a blow up of P² at r distinct points p₁,p₂,, pᵣ. We study lower bounds for Seshadri constants of ample line bundles on Xᵣ. First, we consider the case when p₁,p₂,, pᵣ are on a curve of degree d≤ 3. Then we assume that the points are very general and show that ε(Xᵣ,L,x)≥ 1 for any ample line bundle L and any x∈ Xᵣ if the Strong SHGH conjecture is true. We explore the relation between bounded negativity and Seshadri constants and study the Seshadri function on Xᵣ.