The Hilbert scheme of space curves of small diameter

Abstract

This paper studies space curves C of degree d and arithmetic genus g, with homogeneous ideal I and Rao module M = H 1 * ( Ĩ), whose main results deal with curves which satisfy 0 Ext 2 R (M, M ) = 0 (e.g. of diameter, diam M ≤ 2, which means that M is non-vanishing in at most two consecutive degrees). For such curves C we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, H(d, g), at (C) under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of C turns out to be equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of I. We give classes of obstructed curves C for which we partially compute the equations of the singularity of H(d, g) at (C). Moreover by taking suitable deformations we show how to kill certain repeated direct free factors ("ghost-terms") in the minimal resolution of the ideal of the general curve. For Buchsbaum curves of diameter at most 2, we simplify in this way the minimal resolution further, allowing us to see when a singular point of H(d, g) sits in the intersection of several, or lies in a unique irreducible component of H(d, g). It follows that the graded Betti numbers mentioned above of a generic curve vanish, and that any irreducible component of H(d, g) is reduced (generically smooth) in the diameter 1 case.

The Hilbert scheme of space curves sitting on a smooth surface containing a line

“…A classical way of analyzing whether the closure of a family is a component, possibly nonreduced, is to take a general curve C of a family and describe the curves in an open neighborhood of (C) ∈ H(d, g) sc . More recently several authors has been able to sufficiently describe the obstruction of deforming C and conclude similarly [13,26,32,33,35,36]. In the recent paper [28] we find non-reduced, as well as generically smooth, irreducible components of H(d, g) sc , and we prove non-reducedness along the classical line.…”

“…If X is ACM, or more generally if 0 Ext 2 R (I(X), I(X)) = 0 and X is of maximal rank, then X is unobstructed, and we may replace h 0 (N X ) by dim (X) H(d, g) in (11) (cf. [26,Thm. 2.6], [11]).…”

Families of Artinian and one-dimensional algebras

“…Together with co-authors we have in several papers ( [30], [32], [34], [35], [36] and [33] which makes a correction to [36,Chapter 10]) studied the scheme GradAlg H (R) and its subset PGor(H ) which parametrizes Gorenstein quotients. The latter is essentially an open subscheme of the former (cf.…”

“…In [35] we also considered the minimal resolution of B as well as the minimal resolution of the general elements B 1 and B 2 of W 1 and W 2 , respectively. Indeed…”

“…Indeed (B) sits in the intersection of two irreducible components W 1 and W 2 , both of dimension 4d = 132, of the postulation Hilbert scheme of space curves GradAlg(H B ), cf. [35,Example 35].…”