Narrow Results

We generalize a result of Flenner, proved in characteristic zero, to positive characteristics. We prove that the first syzygy bundle, , of the line bundle over is semistable, for a certain infinite set of integers d ≥ 0. Moreover, for arbitrary d, there is a "good enough estimate" on in terms of d and n; thus a strong restriction theorem of Langer, proved earlier for characteristic k > d, is valid in arbitrary characteristics.

We find sharp bounds on h⁰(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ², we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.