3,680 research outputs found
The modular variety of hyperelliptic curves of genus three
The modular variety of non singular and complete hyperelliptic curves with
level-two structure of genus 3 is a 5-dimensional quasi projective variety
which admits several standard compactifications. The first one, X, comes from
the realization of this variety as a sub-variety of the Siegel modular variety
of level two and genus three .We will be to describe the equations of X in a
suitable projective embedding and its Hilbert function. It will turn out that
X is normal. A further model comes from geometric invariant theory using
so-called semistable degenerated point configurations in (P^1)^8 . We denote
this GIT-compactification by Y. The equations of this variety in a suitable
projective embedding are known. This variety also can by identified with a
Baily-Borel compactified ball-quotient. We will describe these results in some
detail and obtain new proofs including some finer results for them. We have a
birational map between Y and X . In this paper we use the fact that there are
graded algebras (closely related to algebras of modular forms) A,B such that
X=proj(A) and Y=proj(B). This homomorphism rests on the theory of Thomae (19th
century), in which the thetanullwerte of hyperelliptic curves have been
computed. Using the explicit equations for we can compute the base locus
of the map from Y to X.
Blowing up the base locus and the singularity of Y, we get a dominant, smooth
model {\tilde Y}. We will see that {\tilde Y} is isomorphic to the
compactification of families of marked projective lines (P^1,x_1,...,x_8),
usually denoted by {\bar M_{0,8}}. There are several combinatorial similarities
between the models X and Y. These similarities can be described best, if one
uses the ball-model to describe Y.Comment: 39 page
Some Siegel threefolds with a Calabi-Yau model II
In the paper [FSM] we described some Siegel modular threefolds which admit a
Calabi-Yau model. Using a different method we give in this paper an enlarged
list of such varieties that admits a Calabi-Yau model in the following weak
sense: there exists a desingularization in the category of complex spaces of
the Satake compactification which admits a holomorphic three-form without zeros
and whose first Betti number vanishes Basic for our method is the paper [GN] of
van Geemen and Nygaard.Comment: 23 pages, no figure
The Supernova Remnant G11.2-0.3 and its central Pulsar
The plerion inside the composite Supernova Remnant G11.2-0.3 appears to be
dominated by the magnetic field to an extent unprecedented among well known
cases. We discuss its evolution as determined by a central pulsar and the
interaction with the surrounding thermal remnant, which in turn interacts with
the ambient medium. We find that a plausible scenario exists, where all the
observations can be reproduced with rather typical values for the parameters of
the system; we also obtain the most likely period for the still undetected
pulsar.Comment: 10 pages, to be published on ApJ Letters. Formatted using AASTe
Classical theta constants vs. lattice theta series, and super string partition functions
Recently, various possible expressions for the vacuum-to-vacuum superstring
amplitudes has been proposed at genus . To compare the different
proposals, here we will present a careful analysis of the comparison between
the two main technical tools adopted to realize the proposals: the classical
theta constants and the lattice theta series. We compute the relevant Fourier
coefficients in order to relate the two spaces. We will prove the equivalence
up to genus 4. In genus five we will show that the solutions are equivalent
modulo the Schottky form and coincide if we impose the vanishing of the
cosmological constant.Comment: 21 page
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