  The supermeasure whose integral is the genus $g$ vacuum amplitude of
superstring theory is potentially singular on the locus in the moduli space of
supercurves where the corresponding even theta-characteristic has nontrivial
sections. We show that the supermeasure is actually regular for $g\leq 11$. The
result relies on the study of the superperiod map. We also show that the
minimal power of the classical Schottky ideal that annihilates the image of the
superperiod map is equal to $g$ if $g$ is odd and is equal to $g$ or $g-1$ if
$g$ is even.
