Hyperelliptic limits of hypersurfaces through canonical curves and ribbons

Abstract:

We describe explicitly all hyperelliptic limits of hypersurfaces through smooth canonical curves of genus $g$ in $\mathbb {P}^{g-1}$. Also, we construct an open embedding of the blow-up of a $PGL_g$-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 [Trans. Amer. Math. Soc. 347 (1995), pp. 719–756].