Families of Artinian and one-dimensional algebras

The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring $R$ with a special look to level algebras. Let \GradAlg^H(R) be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. Let $B \to A$ be any graded surjection of quotients of $R$ with Hilbert function $H_B$ and $H_A$, and h-vectors $h_B=(1,h_1,...,h_j,...)$ and $h_A$, respectively. If \depth A = \dim A \leq 1 and $A$ is a ``truncation'' of $B$ in the sense that $h_A=(1,h_1,...,h_{j-1},\alpha,0,0,...)$ for some $\alpha \leq h_j$, then we show there is a close relationship between \GradAlg^{H_A}(R) and \GradAlg^{H_B}(R) concerning e.g. smoothness and dimension at the points $(A)$ and $(B)$ respectively, provided $B$ is a complete intersection or provided the Castelnuovo-Mumford regularity of $A$ is at least 3 (sometimes 2) larger than the regularity of $B$. In the complete intersection case we generalize this relationship to ``non-truncated'' Artinian algebras $A$ which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of \GradAlg^H(R), $H=(1,3,6,10,14,10,6,2)$, whose general elements are Artinian level algebras of type 2.Comment: 29 page

Families of artinian and low dimensional determinantal rings

Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous of degrees a_j-b_i. In this paper we extend previous results on the dimension and codimension of W_s(\underline{b};\underline{a}) in GradAlg(H) to {\it artinian determinantal rings}, and we show that GradAlg(H) is generically smooth along W_s(\underline{b};\underline{a}) under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of P^n is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of GradAlg(H).Comment: Postprint replacing preprint. 29 pages. Online 26.May 2017 in Journal of Pure and Applied Algebr

The Hilbert Scheme of Buchsbaum space curves

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H_{*}^0(\sI_C) and Rao module M:=H_{*}^1(\sI_C). By taking suitable generizations (deformations to a more general curve) C' of C, we simplify the minimal free resolution of I(C) by e.g. making consecutive free summands (ghost-terms) disappear in a free resolution of I(C'). Using this for Buchsbaum curves of diameter one (M_v \ne 0 for only one v), we establish a one-to-one correspondence between the set \sS of irreducible components of H(d,g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of \sS), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.Comment: Minor changes in Thm. 6.1 where the particular case (v) is corrected (this inaccuracy occurs also in the published version in Annales de l'institut Fourier, 2012); 23 page

Unobstructedness and dimension of families of Gorenstein algebras

The goal of this paper is to develop tools to study maximal families of Gorenstein quotients A of a polynomial ring R. We prove a very general Theorem on deformations of the homogeneous coordinate ring of a scheme Proj(A) which is defined as the degeneracy locus of a regular section of the dual of some sheaf M^~ of rank r supported on say an arithmetically Cohen-Macaulay subscheme Proj(B) of Proj(R). Under certain conditions (notably; M maximally Cohen-Macaulay and the top exterior power of M^~ a twist of the canonical sheaf), then A is Gorenstein, and under additional assumptions, we show the unobstructedness of A and we give an explicit formula the dimension of any maximal family of Gorenstein quotients of R with fixed Hilbert function obtained by a regular section as above. The theorem also applies to Artinian quotients A. The case where M itself is a twist of the canonical module (r=1) was studied in a previous paper, while this paper concentrates on other low rank cases, notably r=2 and 3. In these cases regular sections of the first Koszul homology module and of normal sheaves to licci schemes (of say codimension 2) lead to Gorenstein quotients (of e.g. codimension 4) whose parameter spaces we examine. Our main applications are for Gorenstein quotients of codimension 4 of R since our assumptions are almost always satisfied in this case. Special attention are paid to arithmetically Gorenstein curves in P^5.Comment: 34 pages. In the 1st version on the arXiv, as well as in the published version in Collect. Math. 58, 2 (2007), there is a missing assumption of generality in the part of the results which deals with the codimension of a stratum (see Rem. 16(ii) and the text before Thm. 15

The Hilbert Scheme of Space Curves of Small Diameter

This paper studies space curves C of degree d and arithmetic genus g, with homogeneous ideal I and Rao module M = H_{*}^1(I^~), whose main results deal with curves which satisfy Ext^2(M,M)_0=0 (e.g. of diameter, diam M < 3, 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 products of 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 products of 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.Comment: Prop. 32(b) is now changed to what is stated in the published version. In Prop. 32(a) we replace "minimal free resolution" by "free resolution which is minimal except possibly in degree t+3", a change that unfortunately is not done in the published version! A sentence in its proof is added. There are changes just before Prop. 24 to credit Fl{\o}ystad, and several changes on the "topos" leve

Families of low dimensional determinantal schemes

A scheme X \subset \PP^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t x t minors of a homogeneous t x (t+c-1) matrix (f_{ij}). Given integers a_0 \le a_1 \le ...\le a_{t+c-2} and b_1 \le ...\le b_t, we denote by W_s(b;a) \subset Hilb(\PP^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_j-b_i and Hilb(\PP^{n}) is the Hilbert scheme (if n-c > 0, resp. the postulation Hilbert scheme if n-c = 0). Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of W_s(b;a) in Hilb(\PP^{n}) and we show that Hilb(\PP^{n}) is generically smooth along W_s(b;a) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dim W_s(b;a) appearing in [26].Comment: 22 page

Comparison theorems for deformation functors via invariant theory

We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.Comment: Minor improvements and corrections. Improved presentation. Changed some definitions and term

Families of determinantal schemes

Given integers a_0 \le a_1 \le ... \le a_{t+c-2} and b_1 \le ... \le b_t, we denote by W(b;a) \subset Hilb^p(\PP^{n}) the locus of good determinantal schemes X \subset \PP^{n} of codimension c defined by the maximal minors of a t x (t+c-1) homogeneous matrix with entries homogeneous polynomials of degree a_j-b_i. The goal of this short note is to extend and complete the results given by the authors in [10] and determine under weakened numerical assumptions the dimension of W(b;a), as well as whether the closure of W(b;a) is a generically smooth irreducible component of the Hilbert scheme Hilb^p(\PP^{n}).Comment: The non-emptiness of W(b;a) is restated as (2.2) in this version; the codimension c=2 case in (2.5)-(2.6) is reconsidered, and c > 2 (c > 3) is now an assumption in (2.16)-(2.17). 13 page