Chern classes and cohomology for rank 2 reflexive sheaves on P3

Abstract

The paper shows that, if F is a nonsplit rank 2 reflexive sheaf on P 3, then the knowledge of the numbers dn = h2(F(n)) - h\F(n)) gives an explicit algorithm to compute the Chern classes C\, Cι, c?> and the dimensions h°(F(n)), for all n (in particular the first integer a such that the sheaf F(a) has some nonzero section). If the sheaf is a vector bundle it is also proved that the knowledge of the numerical sequence {hι(F(n))} together with the first Chern class gives all the information as above. In some special cases, i.e. when hx(F(n)) Φ 0 for at most three values of n, an algorithm is also produced to compute the first Chern class from the sequence {hι(F(n))} . Vector bundles with natural cohomology are also discussed. It must be remarked that, if one knows not only the dimensions hι (F(n)), for all n, but also the whole structure of the Rao-module Q)Hι(F(n)), then the first Chern class C is uniquely determined (as it is shown in a paper by P. Rao).