Splitting type, global sections and Chern classes for torsion free sheaves on PN

In this paper we compare a torsion free sheaf F on PN and the free vector bundle having same rank and splitting type. We show that the first one has always "less" global sections, while it hasa higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of F. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes ci(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove a following formula for the discriminant of any rank n torsion free sheaf on PN, whose splitting type has no gap. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes ci(F(t)), i>2, for the dimension of the cohomology modules HiF(t) and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend on c1(F), c2(F), the splitting type of F and t.