2,955 research outputs found
Hyperelliptic Integrable Systems on K3 and Rational Surfaces
We show several examples of integrable systems related to special K3 and
rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double
covering of the projective plane, a rational elliptic surface, etc.). The
construction, based on Beauvilles's general idea, is considerably simplified by
the fact that all examples are described by hyperelliptic curves and Jacobians.
This also enables to compare these integrable systems with more classical
integrable systems, such as the Neumann system and the periodic Toda chain,
which are also associated with rational surfaces. A delicate difference between
the cases of K3 and of rational surfaces is pointed out therein.Comment: LaTeX2e using packages "amsmath,amssymb", 15 pages, no figur
Linear manifolds in the moduli space of one-forms
We study closures of GL_2(R)-orbits on the total space of the Hodge bundle
over the moduli space of curves under the assumption that they are algebraic
manifolds.
We show that, in the generic stratum, such manifolds are the whole stratum,
the hyperelliptic locus or parameterize curves whose Jacobian has additional
endomorphisms. This follows from a cohomological description of the tangent
bundle to strata. For non-generic strata similar results can be shown by a
case-by-case inspection.
We also propose to study a notion of 'linear manifold' that comprises
Teichmueller curves, Hilbert modular surfaces and the ball quotients of Deligne
and Mostow. Moreover, we give an explanation for the difference between Hilbert
modular surfaces and Hilbert modular threefolds with respect to this notion of
linearity.Comment: Minor modifications, many typos fixe
Relative Prym varieties associated to the double cover of an Enriques surface
Given an Enriques surface T , its universal K3 cover f : S → T , and a genus g linear system |C| on T, we construct the relative Prym variety PH = Prymv,H(D/C), where C → |C| and D → |f∗C| are the universal families, v is the Mukai vector (0, [D], 2−2g) and H is a polarization on S. The relative Prym variety is a (2g−2)-dimensional possibly singular variety, whose smooth locus is endowed with a hyperk ̈ahler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space Mv,H (S). There is a natural Lagrangian fibration η : PH → |C|, that makes the regular locus of PH into an integrable system whose general fiber is a (g − 1)-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if |C| is a hyperelliptic linear system, then PH admits a symplectic resolution which is birational to a hyperk ̈ahler manifold of K3[g−1]-type, while if |C| is not hyperelliptic, then PH admits no symplectic resolution. We also prove that any resolution of PH is simply connected and, when g is odd, any resolution of PH has h2,0-Hodge number equal to one
The arithmetic of Prym varieties in genus 3
Given a curve of genus 3 with an unramified double cover, we give an explicit
description of the associated Prym-variety. We also describe how an unramified
double cover of a non-hyperelliptic genus 3 curve can be mapped into the
Jacobian of a curve of genus 2 over its field of definition and how this can be
used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5
with an unramified involution. As an application, we determine the rational
points on a smooth plane quartic with no particular geometric properties and
give examples of curves of genus 3 and 5 violating the Hasse-principle. We also
show how these constructions can be used to design smooth plane quartics with
specific arithmetic properties. As an example, we give a smooth plane quartic
with all 28 bitangents defined over Q(t).Comment: 21 page
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