Some questions in algebraic geometry (vector bundles, normal bundles and fat points)

In this paper we deal with three argument. In the first part we study rank two globally generated vector bundles on P^n. We classify this bundles through their Chern class, until c_1 = 5 (where c_1 indicates first Chern class). In the second part we deal with normal bundle of projective normal curves. More precisely there is a conjecture, due to Hartshorne and we prove it for some particular case. In the end we study subschemes of P² with fat points. In particulary given Z subscheme of P², we want to understand if Z has maximum rank. We analyze the cases with ten fat points of multiplicity almost eight. To sole this problem, we present a proof of Harbourne-Hirshowitz conjecture for linear system with multiple points of order eight or less and then we prove that every analyzed case has maximum rank.