On the variety associated to the ring of theta constants in genus 3

Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ theta constants of first kind $$ \sum_{n\;{\rm integral}}\exp\pi{\rm i}\big(Z[n+a/2]+2b’(n+a/2)\big),\quad a,b\ {\rm integral}. $$ define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)= \overline{\scr{H}_g/\Gamma_g[4,8]}$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$ X_3(4,8)\longrightarrow\Bbb{P}^{35} $$ is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of $\Bbb{P}^{35}$.