分野別セミナー

Given a projective scheme V, let Hilb^{sc} V denote the Hilbert scheme of smooth connected curves in V. Mumford first proved that Hilb^{sc} P^3 contains a generically non-reduced (irreducible) component as a pathology of the Hilbert scheme. Later, many examples of such non-reduced components (of Hilb^{sc} P^3) were found by many algebraic geometers, e.g.~ Kleppe, Ellia, Gruson-Peskine, Floystad, Kleppe-Ottem, etc. Recently, Mukai and the speaker [MN] have generalized Mumford’s example and have proved that for many uniruled 3-folds V, Hilb^{sc} V contains infinitely many generically non-reduced components.

In this talk, we discuss the deformations of a smooth curve C on a smooth projective 3-fold V, assuming the presence of a smooth (intermediate) surface S satisfying $C \subset S \subset V$. Generalizing a result in [MN], we give a new sufficient condition for a first order infinitesimal deformation of C in V to be primarily obstructed. In particular, when V is Fano and S is K3, we give a sufficient condition for C to be (un)obstructed in V, in terms of (-2)-curves and elliptic curves on S. Applying this result, we prove that the Hilbert scheme Hilb^{sc} V_4 of a smooth quartic 3-fold V_4 contains infinitely many generically non-reduced components. This talk is based on arXiv:1601.07301.