  In this paper, we study the fattening effect of points over the complex
numbers for del Pezzo surfaces $\mathbb{S}_r$ arising by blowing-up of
$\mathbb{P}^2$ at $r$ general points, with $ r \in \{1, \dots, 8 \}$. Basic
questions when studying the problem of points fattening on an arbitrary variety
are what is the minimal growth of the initial sequence and how are the sets on
which this minimal growth happens characterized geometrically. We provide
complete answer for del Pezzo surfaces.
