Hyperelliptic limits of quadrics through canonical curves and ribbons

TL;DR

This paper explicitly describes hyperelliptic limits of quadrics through smooth canonical curves and constructs a geometric embedding relating moduli spaces and Hilbert schemes, advancing understanding of degenerations in algebraic geometry.

Contribution

It provides explicit descriptions of hyperelliptic limits of quadrics and constructs a new embedding connecting moduli spaces with Hilbert schemes involving canonical ribbons.

Findings

Explicit description of hyperelliptic limits of quadrics

Construction of an open embedding between moduli and Hilbert schemes

Analysis of canonical ribbons as degenerations of rational normal curves

Abstract

We describe explicitly all hyperelliptic limits of quadrics through smooth canonical curves of genus $g$ in ${\mathbb P}^{g-1}$. Also, we construct an open embedding of the blow up of a ${\rm PGL}_g$-bundle over the moduli space of curves of genus $g$ along the hyperelliptic locus into the blow up of the canonical Hilbert scheme of ${\mathbb P}^{g-1}$ along the closure of the locus of canonical ribbons, which are certain double thickenings of rational normal curves introduced and studied by Bayer and Eisenbud.