2025/01/27 by Nasu, Hirokazu
#14C05 #14D15 #14H50 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.15788
We study the deformations of space curves C ⊂ \mathbb P4, assuming that they are contained in a smooth complete intersection S2,2 ⊂ \mathbb P4, 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≥ 2d-12, and h1(C,\mathcal IC(2))=1, then C is obstructed and stably degenerate, i.e., C has some first order infinitesimal deformations in \mathbb P4 not contained in any deformations of S2,2 in \mathbb P4, but they do not lift to any global deformations. (As a result, every global deformation of C in \mathbb P4 is contained in a deformation of S2,2 in \mathbb P4.) As an application, we construct infinitely many examples of irreducible components of the Hilbert scheme Hilbsc \mathbb P4 of smooth connected curves in \mathbb P4, along which Hilbsc \mathbb P4 is generically non-reduced. In the case d=14 and g=16, we obtain a non-reduced component of Hilbsc \mathbb P4 of dimension 55 with dim THilbsc \mathbb P4=57, analogous to Mumford's example of a non-reduced component of Hilbsc \mathbb P3, whose general member is contained in a smooth cubic surface S3 ⊂ \mathbb P3.