On generically non-reduced components of Hilbert schemes of smooth curves
Ananyo Dan
TL;DR
The paper tackles the existence of generically non-reduced components in Hilbert schemes of smooth curves in $\mathbb{P}^3$ by integrating Hodge-theoretic methods with flag Hilbert schemes. It develops a general criterion tying non-reducedness of Hodge loci to non-reduced flag-Hilbert components via the Gauss–Manin connection and the semi-regularity map, and proves that certain Hodge loci are generically non-reduced. Through explicit constructions (notably using Martin-Deschamps–Perrin type divisors) and CM-regularity arguments, it identifies concrete cases where the Hodge locus is generically non-reduced and lifts these to non-reduced components of the Hilbert scheme of curves. The main result shows that for $d\ge 5$ and $m\ge 2d-2$, there exist generically non-reduced irreducible components of the Hilbert scheme parameterizing smooth curves of degree $md+3$ with a prescribed genus, lying on smooth degree-$d$ surfaces with Picard number at least $3$ and not contained in smaller-degree surfaces, thereby extending Mumford’s phenomenon to higher-degree and higher Picard-number settings.
Abstract
A classical example of Mumford gives a generically non-reduced component of the Hilbert scheme of smooth curves in the projective 3-space such that a general element of the component is contained in a smooth cubic hypersurface in the projective 3-space. In this article we use techniques from Hodge theory to give further examples of such (generically non-reduced) components of Hilbert schemes of smooth curves without any restriction on the degree of the hypersurface containing it. As a byproduct we also obtain generically non-reduced components of certain Hodge loci.