The modular variety of hyperelliptic curves of genus three
Abstract
The modular variety of non singular and complete hyperelliptic curves with leveltwo structure of genus 3 is a 5dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a subvariety 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 socalled semistable degenerated point configurations in (P^1)^8 . We denote this GITcompactification by Y. The equations of this variety in a suitable projective embedding are known. This variety also can by identified with a BailyBorel compactified ballquotient. 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 $A,B$ 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 ballmodel to describe Y.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.5920
 Bibcode:
 2007arXiv0710.5920F
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra
 EPrint:
 39 pages