  Combining certain identities for modular forms due to Igusa with
Schottky-Jung relations, we study the cosmological constant for the recently
proposed ansatz for the chiral superstring measure in genus 5. The vanishing of
this cosmological constant turns out to be equivalent to the long-conjectured
vanishing of a certain explicit modular form of genus 5 on the moduli of curves
M_5, and we disprove this conjecture, thus showing that the cosmological
constant for the proposed ansatz does not vanish identically. We exhibit an
easy modification of the genus 5 ansatz satisfying factorization constraints
and yielding a vanishing cosmological constant. We also give an expression for
the cosmological constant for the proposed ansatz that should hold for any
genus if certain generalized Schottky-Jung identities hold.
