We discuss the recently proposed ansatze for the chiral superstring measure, as the product of the bosonic measure and a suitable modular form. We explain how this modular form can be constructed from factorization constraints (restrictions to lower genera), as done by D'Hoker and Phong in their work on the two-loop superstring amplitudes. We further show that the cosmological constant for the proposed ansatz vanishes, as expected, for genera up to 5, and discuss the computation of N-point amplitudes and the results on their vanishing for N<4.