Singularities of the Hilbert scheme of effective divisors

Abstract

In this article, we study the Hilbert scheme of effective divisors in smooth hypersurfaces in $\mathbb{P}^3$, a topic not extensively studied. We prove that there exists such effective divisors $D$ satisfying the property: there exists infinitesimal deformations of $D$ not deforming the associated reduced scheme $D_{\mathrm{red}}$. We observe that such infinitesimal deformations contribute to non-reducedness of the corresponding Hilbert scheme. We finally introduce a concept of simple extension of curves and notice that the above mentioned property is preserved under simple extension of curves.