What is missing in canonical models for proper normal algebraic surfaces?

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.Comment: 10 pages, minor changes, to appear in Abh. Math. Sem. Univ. Hambur

Remarks on the existence of Cartier divisors

Given an invertible sheaf, does it come from a Cartier divisor? This might fail in presence of embedded components. I give some examples and characterize those invertible sheaves that allow a Cartier divisor.Comment: 5 pages, to appear in Arch. Mat

On non-projective normal surfaces

We construct examples of non-projective normal proper algebraic surfaces and discuss the pathological behaviour of their Neron-Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus.Comment: 7 pages, Latex2e, to be published in Manuscripta Mathematic

The T^1-lifting theorem in positive characteristic

Replacing symmetric powers by divided powers and working over Witt vectors instead of ground fields, I generalize Kawamatas T^1-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 cohomology modules.Comment: 13 pages, minor changes, to appear in J. Algebraic Geo

The strong Franchetta Conjecture in arbitrary characteristics

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>2. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.Comment: 23 pages, major extension, to appear in Internat. J. Mat

Singularities appearing on generic fibers of morphisms between smooth schemes

I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.Comment: 20 page

On contractible curves on normal surfaces

We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on homogeneous spectra defined by Weil divisors on surfaces.Comment: 20 pages, submitted to J. Reine Angew. Mat

Effective descent maps for schemes

The paper is withdrawn due to mistakes in the proofs for Proposition 1.2 and Theorem 2.2

Kummer surfaces for the selfproduct 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 in 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 after purely inseparable base change.Comment: 33 pages, 8 figures. Artin invariants correcte

There are enough Azumaya algebras on surfaces

Using Maruyama's theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces.Comment: 13 pages, major revision, to appear in Math. An