242 research outputs found
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
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
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
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