On generically non-reduced components of Hilbert schemes of smooth curves

TL;DR

This paper uses Hodge theory to identify new examples of non-reduced components in Hilbert schemes of smooth curves, expanding on classical examples and exploring their relation to Hodge loci.

Contribution

It introduces novel techniques to find non-reduced components of Hilbert schemes of smooth curves without degree restrictions, extending classical results.

Findings

Identification of new non-reduced components in Hilbert schemes

Examples of non-reduced components in Hodge loci

Extension of classical Mumford's example

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.