Almost Regular Bundles on del Pezzo Fibrations

Abstract

This paper is devoted to the study of a certain class of principal bundles on del Pezzo surfaces, which were introduced and studied by Friedman and Morgan in \cite{FMdP}: The two authors showed that there exists a unique principal bundle (up to isomorphism) on a given (Gorenstein) del Pezzo surface satisfying certain properties. We call these bundles {\em almost regular}. In turn, we study them in families. In this case, the existence and the moduli of these bundles are governed by the cohomology groups of an abelian sheaf ${\mathscr A}$: On a given del Pezzo fibration, the existence of an almost regular bundle depends on the vanishing of an obstruction class in $H^2({\mathscr A})$. In which case, the set of isomorphism classes of almost regular bundles become a homogeneous space under the $H^1({\mathscr A})$ action.