  There are two main problems in finding the higher genus superstring measure.
The first one is that for $g\geq 5$ the super moduli space is not projected.
Furthermore, the supermeasure is regular for $g\leq 11$, a bound related to the
source of singularities due to the divisor in the moduli space of Riemann
surfaces with even spin structure having holomorphic sections, such a divisor
is called the $\theta$-null divisor. A result of this paper is the
characterization of such a divisor. This is done by first extending the Dirac
propagator, that is the Szeg\"o kernel, to the case of an arbitrary number of
zero modes, that leads to a modification of the Fay trisecant identity, where
the determinant of the Dirac propagators is replaced by the product of two
determinants of the Dirac zero modes. By taking suitable limits of points on
the Riemann surface, this {\it holomorphic Fay trisecant identity} leads to
identities that include points dependent rank 3 quadrics in $\mathbb{P}^{g-1}$.
Furthermore, integrating over the homological cycles gives relations for the
Riemann period matrix which are satisfied in the presence of Dirac zero modes.
Such identities characterize the $\theta$-null divisor. Finally, we provide the
geometrical interpretation of the above points dependent quadrics and shows,
via a new $\theta$-identity, its relation with the Andreotti-Mayer quadric.
