Obstructions to deforming space curves lying on a del Pezzo surface

Abstract: We study the deformations of space curves $C \subset \mathbb P^4$, assuming that they are contained in a smooth complete intersection $S_{2,2} \subset \mathbb P^4$, i.e., a smooth del Pezzo surface of degree $4$. We give sufficient conditions for $C$ to be (un)obstructed in terms of the degree $d$ and the genus $g$ of $C$. We prove that if $d>8$, $g\ge 2d-12$, and $h^1(C,\mathcal I_C(2))=1$, then $C$ is obstructed and stably degenerate, i.e., $C$ has some first order infinitesimal deformations in $\mathbb P^4$ not contained in any deformations of $S_{2,2}$ in $\mathbb P^4$, but they do not lift to any global deformations. (As a result, every global deformation of $C$ in $\mathbb P^4$ is contained in a deformation of $S_{2,2}$ in $\mathbb P^4$.) As an application, we construct infinitely many examples of irreducible components of the Hilbert scheme $\operatorname{Hilb}^{sc} \mathbb P^4$ of smooth connected curves in $\mathbb P^4$, along which $\operatorname{Hilb}^{sc} \mathbb P^4$ is generically non-reduced. In the case $d=14$ and $g=16$, we obtain a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^4$ of dimension $55$ with $\dim T_{\operatorname{Hilb}^{sc} \mathbb P^4}=57$, analogous to Mumford's example of a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^3$, whose general member is contained in a smooth cubic surface $S_3 \subset \mathbb P^3$.