Stefan Schröer

Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors

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-Baillys pencil of abelian surfaces and Katsuras analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.

The Brauer group of analytic K3 surfaces

We show that for complex analytic K3 surfaces any torsion class in H2(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 then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family. In analogy to the isomorphism Pic(X) ∼= H1(X,O∗ X), Grothendieck investigated in [8] the possibility of interpreting classes in H2(X,O∗ X) as geometric objects. He observed that the Brauer group Br(X), parameterizing equivalence classes of sheaves of Azumaya algebras on X, naturally injects into H2(X,O∗ X). It is not difficult to see that Br(X) ⊂ H2(X,O∗ X) is contained in the torsion part of H2(X,O∗ X) and Grothendieck asked: Is the natural injection Br(X ) ⊂ H (X ,O∗ X )tor an isomorphism? This question is of interest in various geometric categories, e.g. X might be a scheme, a complex space, a complex manifold, etc. It is also related to more recent developments in the application of complex algebraic geometry to conformal field theory. Certain elements in H2(X,O∗ X) have been interpreted as so-called B-fields, and those are used to construct super conformal field theories associated to Ricci-flat manifolds. Thus, understanding the geometric meaning of the cohomological Brauer group Br′(X) := H2(X,O∗ X)tor is also of interest for the mathematical interpretation of string theory and mirror symmetry. An affirmative answer to Grothendieck’s question has been given only in very few special cases: • If X is a complex curve, then H2(X,O∗ X) = 0. Hence, Br(X) = Br ′(X) = H2(X,O∗ X) = 0 (see [8, Cor.2.2] for the general case of a curve). • For smooth algebraic surfaces the surjectivity has been proved by Grothendieck [8, Cor.2.2] and for normal algebraic surfaces a proof was given more recently by Schroer [14]. • Hoobler [9] and Berkovich [3] gave an affirmative answer for abelian varieties of any dimension and Elencwajg and Narasimhan gave another proof for complex tori [6].

Kummer surfaces for the self-product of the cuspidal rational curve

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 actions, I give the correct Kummer-type construction for this situation. We encounter rational double points of type D4 and D8, instead of type A1. 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.

ON GENUS CHANGE IN ALGEBRAIC CURVES OVER IMPERFECT FIELDS

We give a new proof, in scheme-theoretic language, of Tates classical result on genus change of curves over imperfect fields in characteristic p > 0. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus over the algebraic closure is divisible by (p-1)/2.

Homogeneous Coordinates and Quotient Presentations for Toric Varieties

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas ℚ–Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.

On embeddings into toric prevarieties

We give examples of complete normal surfaces that are not em- beddable into simplicial toric prevarieties nor toric prevarieties of ane inter- section.

Irreducible Degenerations of Primary Kodaira Surfaces

We classify irreducible d-semistable degenerations of primary Kodaira surfaces. As an application we construct a canonical partial completion for the moduli space of primary Kodaira surfaces.

On Fibrations whose geometric fibers are nonreduced

We give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus 1 curves.

Geometry on totally separably closed schemes

We prove, for quasicompact separated schemes over ground fields, that Čech cohomology coincides with sheaf cohomology with respect to the Nisnevich topology. This is a partial generalization of Artin’s result that for noetherian schemes such an equality holds with respect to the étale topology, under the somewhat unnatural assumption that every finite subset admits an affine open neighborhood (AF-property). Our key result is that on the absolute integral closure of separated algebraic schemes, the intersection of any two irreducible closed subsets remains irreducible. We prove this by establishing general modification and contraction results adapted to inverse limits of schemes. Along the way, we characterize schemes that are acyclic with respect to various Grothendieck topologies, study schemes all whose local rings are strictly henselian, and analyze fiber products of strict localizations.

Baer's Result: The Infinite Product of the Integers Has No Basis

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