575 research outputs found
Second order theta divisors on Pryms
Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed
characterizations of the locus of jacobians which use the linear system of
-divisors. We give new evidence for these conjectures in the case of
Prym varieties.Comment: AMS-Latex, 11 pages, the exposition has been modified, Proposition 5
has been modifie
A Prym construction for the cohomology of a cubic hypersurface
Mumford defined a natural isomorphism between the intermediate jacobian of a
conic-bundle over and the Prym variety of a naturally defined \'etale
double cover of the discrminant curve of the conic-bundle. Clemens and
Griffiths used this isomorphism to give a proof of the irrationality of a
smooth cubic threefold and Beauville later generalized the isomorphism to
intermediate jacobians of odd-dimensional quadric-bundles over . We
further generalize the isomorphism to the primitive cohomology of a smooth
cubic hypersurface in .Comment: AMS-Latex, 39 page
Density and completeness of subvarieties of moduli spaces of curves or abelian varieties
Let be a subvariety of codimension of the moduli space \cA_g
of principally polarized abelian varieties of dimension or of the moduli
space \tM_g of curves of compact type of genus . We prove that the set
of elements of which map onto an elliptic curve is analytically
dense in . From this we deduce that if V \subset \cA_g is complete, then
has codimension equal to and the set of elements of isogenous to a
product of elliptic curves is countable and analytically dense in . We
also prove a technical property of the conormal sheaf of if V \subset
\tM_g (or \cA_g) is complete of codimension .Comment: AMS-LaTeX, 15 page
Deforming curves in jacobians to non-jacobians I: curves in
We introduce deformation theoretic methods for determining when a curve
in a non-hyperelliptic jacobian will deform with to a non-jacobian.
We apply these methods to a particular class of curves in the second symmetric
power of . More precisely, given a pencil of degree on
, let be the curve parametrizing pairs of points in divisors of
(see the paper for the precise scheme-theoretical definition). We prove that if
deforms infinitesimally out of the jacobian locus with then either
or , dim and has genus 4.Comment: amslatex, 25 page
An inductive approach to the Hodge conjecture for abelian varieties
We describe an inductive approach most appropriate for abelian varieties with
an action of an imaginary quadratic field.Comment: amslatex, 11 page
Some properties of second order theta functions on Prym varieties
Let be the two component Prym variety associated to an \'etale
double cover of a non-hyperelliptic curve of genus
and let and be the linear systems of second order theta
divisors on and respectively. The component contains canonically
the Prym curve . We show that the base locus of the subseries of
divisors containing is scheme-theoretically the curve
. We also prove canonical isomorphisms between some subseries of
and and some subseries of second order theta divisors on
the Jacobian of .Comment: references adde
Ampleness of intersections of translates of theta divisors in an abelian fourfold
We prove that the cotangent bundle of a complete intersection of two general
translates of the theta divisor of the jacobian of a general curve of genus 4
is ample. From this the same result for a general principally polarized abelian
variety of dimension 4 follows.Comment: ams-latex, 8 page
The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian
We complete the proof of the fact that the moduli space of rank two bundles
with trivial determinant embeds into the linear system of divisors on
which are linearly equivalent to . The embedded tangent
space at a semi-stable non-stable bundle , where is a
degree zero line bundle, is shown to consist of those divisors in
which contain where is the translate of
by . We also obtain geometrical results on the structure of this
tangent space.Comment: 33 pages, AMS-Late
The primitive cohomology of the theta divisor of an abelian fivefold
The primitive cohomology of the theta divisor of a principally polarized
abelian variety of dimension is a Hodge structure of level . The Hodge
conjecture predicts that it is contained in the image, under the Abel-Jacobi
map, of the cohomology of a family of curves in the theta divisor. In this
paper we use the Prym map to show that this version of the Hodge conjecture is
true for the theta divisor of a general abelian fivefold.Comment: 59 page
Correspondences with split polynomial equations
We introduce endomorphisms of special jacobians and show that they satisfy
polynomial equations with all integer roots which we compute. The eigen-abelian
varieties for these endomorphisms are generalizations of Prym-Tjurin varieties
and naturally contain special curves representing cohomology classes which are
not expected to be represented by curves in generic abelian varieties.Comment: AMS-Latex, 32 pages, minor corrections, computation of dimensions of
eigen-abelian varieties adde
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