Genus 4 Supermoduli Space Is Not Projected

Abstract

The supermoduli stack ${\mathfrak{M}}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g\ge 5$. We extend this result to the even-spin component of ${\mathfrak{M}}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal{SM}}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak{M}}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak{M}}_4^{+}$ is non-projected.