The rationality of ineffective spin genus-4 theta-null loci

ABSTRACT In this paper we show that the divisor in $S^{+}_g$ given by couples $[C,\theta]$, where C is a curve of genus 4 with a vanishing theta-null and θ is an ineffective theta-characteristic, is a rational variety. Actually we first show that for g = 4, the corresponding theta-null Prym divisor in ${\mathcal{R}}_4$ is rational.