The effect of points fattening on the blow up of the projective plane at one point

Abstrakt

In this paper we study the points fattening effect over the complex numbers for the surface arising by blowing-up of $\mathbb{P}^2$ at one point. We denote this space by $\mathbb{S}_1$. This surface has been recently considered with respect to the points fattening, but as a Hirzebruch surface.~We study this issue for $\mathbb{S}_1$ taken as del Pezzo surface.~The choice of point of view forthis space implies the~choice of reference line bundle.~We will show, among others, that the~choice of the polarization is a fundamental factor affecting the shape of the initial sequence.