Almost Regular Bundles on Del Pezzo Fibrations

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 [10]: 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 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 A : On a given del Pezzo fibration, the existence of an almost regular bundle depends on the vanishing of an obstruction class in H(A ). In which case, the set of isomorphism classes of almost regular bundles become a homogeneous space under the H(A ) action. A del Pezzo surface, S, is a smooth complex projective surface whose anticanonical bundle ω S is nef and big. The classification of del Pezzo surfaces shows that such a surface is either isomorphic to P×P or is the blow-up of P at 0 ≤ r ≤ 8 points in almost general position (position presque-générale [5]). A Gorenstein del Pezzo surface Y is a normal rational projective surface whose anticanonical sheaf ω Y is invertible (=Gorenstein) and ample. Such a surface is the anticanonical model of a del Pezzo surface S. A principal bundle is a generalization of a vector bundle in which the fibers of the bundle, previously copies of a fixed vector space V , are now replaced with the copies of a fixed (complex Lie) group G. In this paper, we study a class of principal bundles on families of del Pezzo surfaces which are “natural” in a certain sense. We will call these bundles almost regular. Friedman and Morgan [10] construct such bundles on a single surface Y and show that they are all isomorphic. In other words, there exists a tautological isomorphism class of bundles on Y . Our question is to study these bundles on a family of (Gorenstein) del Pezzo surfaces, p : Y−→X, where p is a projective integral flat map whose geometric fibers are (Gorenstein) del Pezzo surfaces. We formulate our solution in terms of tools developed for principal bundles on elliptic fibrations and Higgs bundles, such as cameral covers and abstract Higgs bundles, some basic singularity theory (simultaneous resolutions of rational double points), and (sub)regular elements from reductive Lie groups. Our solution, most satisfactory when the base of the family is a curve (Theorem 2.39), shows that almost regular bundles on Y can be classified mainly in terms of maximal torus bundles on a cameral cover X̃ (Definition 1.10) ofX (Theorem 2.37). Such a classification is along the classical lines of spectral covers, where Supported by NSF grants DMS-01004354, FRG-0139799 and Schwerpunktprogramm ”Globale Methoden in der komplexen Geometrie” HU 337/5-2.