The bielliptic locus in the Hilbert scheme of canonical curves is unirational
Andrei Stoenică
TL;DR
The paper proves that the bielliptic locus of canonical curves of genus $g \ge 11$ inside the Hilbert scheme $\operatorname{Hilb}_{(2g-2)t + 1-g, \mathbb{P}^{g-1}}$ is unirational, providing a new route to the known unirationality in the moduli space $M_g$. It develops a framework centered on elliptic normal cones: first, a unirational parameter space $H_c$ for these cones in $\mathbb{P}^{g-1}$, then a projective bundle $\mathcal{P}$ over $H_c$ whose fibers parametrize quadrics not containing the cone. A dominant rational map from $\mathcal{P}$ to the bielliptic locus in the Hilbert scheme is constructed via a universal family and incidence with quadrics, and since $H_c$ is unirational, $\mathcal{P}$ is also unirational, implying the target locus is unirational. The argument closes by noting the natural map to the bielliptic locus in $M_g$ yields unirationality there as well for $g \ge 11$.
Abstract
In this paper we prove the unirationality of the locus of bielliptic curves in the Hilbert scheme of canonical curves of genus $g \geq 11$. As a consequence, we obtain another proof for the unirationality of the bielliptic locus in the moduli space of curves of genus $g \geq 11$.