On the Segre Invariant for rank two vector bundles on $\mathbb{P}$^2

We extend the concept of Segre’s Invariant to vector bundles on a surface $X$. For $X={\mathbb{P}}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern’s classes. The irreducibility of strata with fixed Segre’s invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether’s Theory for rank $2$ vector bundles on ${\mathbb{P}}^2\mathrm{.}$