Obstructions to deforming space curves lying on a del Pezzo surface

TL;DR

This work analyzes deformations of smooth space curves $C\subset\mathbb{P}^4$ that lie on a smooth quartic del Pezzo surface $S=S_{2,2}$. It develops a precise obstruction theory, including exterior components and infinitesimal deformations with poles, to determine when such curves yield smooth or non-smooth behavior in the Hilbert scheme $\operatorname{Hilb}^{sc}\mathbb{P}^4$. The authors prove unobstructedness criteria (e.g., when $H^1(I_C(2))=0$ or $H^1(O_C(2))=0$) and compute dimensions of components, showing the existence of infinitely many non-reduced components; they present a Mumford-like example with $d=14$, $g=16$, and $h^1(I_C(2))=1$ to illustrate the phenomenon. The results extend prior curvature- and cubic-surface deformation theories to higher-dimensional settings, illustrating how obstructions can force non-reducedness in Hilbert schemes of space curves. The approach offers a framework for constructing and analyzing obstructed families of curves contained in special surfaces, with potential implications for broader questions about deformations in projective spaces.

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 $S_3 \subset \mathbb P^3$.

Figures (1)

• Figure 1: region of $(d,g)$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Lemma 2.1
• Definition 2.2
• Lemma 2.3: cf. Kleppe87,Nasu9
• Definition 2.4
• Lemma 2.5: Nasu9
• Definition 2.6
• Lemma 2.7