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 small 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 \(S_3 \subset \mathbb {P}^3\) .
