The modular variety of hyperelliptic curves of genus three

The modular variety of nonsingular 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 realizes this variety as a subvariety of the Siegel modular variety of level two and genus three.

It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components.

Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with $8$ branch points. There are two important compactifications of this configuration space. The first one, $Y$, uses the semistable degenerated point configurations in $(P^1)^8$. This variety also can be identified with a Baily-Borel compactified ball-quotient $Y=\overline{\mathcal{B}/\Gamma[1-{\textrm i}]}.$ We will describe these results in some detail and obtain new proofs including some finer results for them. The other compactification uses the fact that families of marked projective lines can degenerate to stable marked curves of genus 0.

We use the standard notation $\bar M_{0,8}$ for this compactification. We have a diagram

$$\xymatrix{ &\bar M_{0,8}\ar[dl]\ar[dr]&\\ Y\ar@{-->}[rr]& &X\;.}$$

The horizontal arrow is only birational but not everywhere regular.

In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) $A,B$ such that $X=\mathop{\rm proj}\nolimits(A), Y=\mathop{\rm proj}\nolimits(B).$