turkmath.org

One of the well-known pathologies in characteristic $p>0$ is non-smoothness of Picard schemes of surfaces. This pathology is closely related to the failure of Kodaira vanishing theorem (KVT) and of Bogomolov inequality; it was conjectured that a modified form of the Bogomolov-Miyaoka-Yau inequality which contains a correction term measuring non-smoothness of the Picard scheme holds for all surfaces in characteristic $p$. In the light of the main result in Jang, it appears that this correction term, if exists, should in fact account for the non-ordinarity of the Picard scheme. It was proved by Mukai that the failure of KVT on a smooth variety $X$ of arbitrary dimension, is equivalent to the existence of a certain purely inseparable finite cover of X of degree $p$. In a related setting, in the work on the failure of Bogomolov's inequality $c_1^2(E) < 4c_2(E)$ for a semi-stable rank two vector bundle $E$ on a surface $X$, one of the key ingredients is the construction of a purely inseparable cover of $X$. This fact provides the motivation for this talk; we address the existence of a special type of inseparable Galois covers, namely the principal homogeneous spaces on $X$ for groups of type $\cal{G}_{a,b}$. These covers contain as a particular case the $\alpha_L$-covers of $X$. We prove the triviality of such covers for suitable pairs $(X, L)$ of a surface $X$ and a line bundle $L$ on $X$. We will conjecture that if $X$ is an ordinary surface, then for any semi-stable vector bundle $E$ on the surface $X$ we have $c_1^2(E) < 4c_2(E)$ by combining the works of Mukai and Shepherd-Barron.