Automorphisms and moduli of surfaces of general type in positive characteristic

In this talk I will discuss the moduli of surfaces of general type defined over an algebraically closed field of characteristic p>0 and the pathologies
that appear compared to the characteristic zero case. In particular, unlike the characteristic zero case, there does not always exists a modular family.
A modular family is a family with the property that up to etale base change, every other family is obtained from it by base change. I will explain the
relation between this pathology and the existence of surfaces of general type with nontrivial global vector fields, or equivalently with nonreduced
automorphism scheme. Moreover, several results regarding the geometry of such surfaces will be presented.