Some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero are constructed. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron-Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus.

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces. 1991 Mathematics Subject Classification. 14F22, 20J06, 32J15.Theorem. Let X be a complex Lie group or a Hopf manifold. Then the inclusion Br(X) ⊂ Br ′ (X) is an equality.This generalizes results of Iversen on characterfree algebraic groups [32], and Hoobler [28], Berkovič [3], and Elencwajg and Narasimhan [16] on abelian varieties and complex tori. Turning to surfaces, we obtain the second main result of this paper:Theorem. Let S be a smooth compact complex-analytic surface with b 1 = 1. Then the inclusion Br(S) ⊂ Br ′ (S) is an equality.Here the main challenge is the case of elliptic surfaces. Usually, such surfaces do not satisfy the required conditions on π 1 (S) and π 2 (S), due to the presence of singular fibers in the elliptic fibration S → B. However, there is always a Zariski open subset U ⊂ S with the desired properties. Some additional arguments then show that this is enough for our purpose. A key ingredient is Hoobler's result [28],

The classical Kummer construction attaches a K3 surface to an abelian surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the self-product of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, we give the correct Kummer-type construction for this situation. We encounter rational double points of type D 4 and D 8 instead of type A 1 . It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change. Contents STEFAN SCHRÖERmaterial on blowing up rational surface singularities along curves. We use this in Section 11, where we consider blowing ups of Weil divisors in genus-one fibrations. This is crucial in Section 12, where we view our normal K3 surfaces as lying in a family Z → S, and show that this family admits a simultaneous resolution X → S after a purely inseparable base change S → S. In the last section we show that this family is induced from a family X → P 1 .

I construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity q > 0. This can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo surfaces, which were classified by Reid. The twisting is with respect to the flat topology and infinitesimal group scheme actions. The twisted surfaces appear as generic fibers for Fano-Mori contractions on certain threefolds with only canonical singularities.