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Journals and Conferences
The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme… (More)
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… (More)
I show that each étale n-cohomology class on noetherian schemes comes from a Čech cocycle, provided that any n-tuple of points admits an affine open neighborhood. Together with results of Raeburn and Taylor on the bigger Brauer group, this implies that for schemes such that each pair of points admits an affine open neighborhood, any étale Gm-gerbe comes… (More)
I construct some smooth Calabi–Yau threefolds in characteristic two and three that do not lift to characteristic zero. 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… (More)
We show that for complex analytic K3 surfaces any torsion class in H(X,O X) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and… (More)
Using Moriwaki’s calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g ≥ 3. Similar results hold for generic pointed curves.
In modern form, Hilbert’s Theorem 90 tells us that Rǫ∗(Gm) = 0, where ǫ : Xét → Xzar is the canonical map between the étale site and the Zariski site of a scheme X. I construct examples showing that the corresponding statement for algebraic spaces does not hold.
I construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity q > 0. Such things 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… (More)
The neem tree produces highly specified acting insecticides mainly in its seeds. By pressurizing or extracting the seeds an insecticide oil can be manufactured. For successful application emulsifiers are needed to render the oil soluble in water. The heavy oil has to be stable in emulsion, but on the other hand the surfactant should not reduce the… (More)
Replacing symmetric powers by divided powers and working over Witt vectors instead of ground fields, I generalize Kawamata’s T -lifting theorem to characteristic p > 0. Combined with the work of Deligne–Illusie on degeneration of the Hodge–de Rham spectral sequences, this gives unobstructedness for certain Calabi–Yau varieties with free crystalline… (More)