The modular variety of hyperelliptic curves of genus three
| dc.creator | Freitag, E. | |
| dc.creator | Manni, R. Salvati | |
| dc.date | 2007-10-31 | |
| dc.date.accessioned | 2026-07-07T08:39:46Z | |
| dc.date.available | 2026-07-07T08:39:46Z | |
| dc.description | 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 $A,B$ 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 ball-model to describe Y. | |
| dc.description | 39 pages | |
| dc.identifier | https://arxiv.org/abs/0710.5920 | |
| dc.identifier | http://arxiv.org/abs/0710.5920 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141186 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.title | The modular variety of hyperelliptic curves of genus three | |
| dc.type | text |