Antonio Lanteri

A class of vector bundles of rank on an ‐dimensional complex projective variety , spanned by global sections, with positive top Chern class and ample determinant is investigated. The standard relation between Chern and Segre polynomials allows us to decide about the bigness of a bundle closely related to . The size , an integer measuring the deviation of from bigness, is introduced and studied. The structure of , in connection with the behavior of the size, yields a useful tool to produce several examples. Among them the pair emerges, of which two characterizations are provided in terms of and in terms of the degree of curves representing with respect to the determinant.

Let [ ] be any Fano manifold polarized by a positive multiple of its fundamental divisor [ ]. The polynomial defining the Hilbert curve of [ ] reduces to the Hilbert polynomial of [ ], hence it is totally reducible over [ ]; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if [ ] has index [ ]. It is natural to ask when the same happens for all linear factors. Here the total reducibility over [ ] of the Hilbert polynomial is investigated for three special kinds of Fano manifolds: Fano manifolds of large index, toric Fano manifolds of low dimension, and projectivized Fano bundles of low coindex.

Polarized rational surfaces ( X , L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽 1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations ( X , L) are classified and a related property concerned with the inflectional locus is discussed.