Recent work of Schenck, Stillman and Yuan arXiv:2011.10871 outlines all possible Betti tables for Artin Gorenstein algebras $A$ with regularity($A$) = 4 = codim($A$). We populate the second half of this list with examples of stable curves, and ask if there are further possible constructions. The problem of deformation between curves with the same Hilbert series but different Betti tables is ongoing work, but our work solves one case: a deformation (due to Jan Stevens) between a reducible curve corresponding to Betti table type 2.7 in arXiv:2011.10871 and the curve obtained as the intersection of a del Pezzo surface of degree 5 and a cubic hypersurface.
Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$ in an attempt to see how far typical codimension~2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. These examples are candidates for counterexamples to the hoped-for extensions of codimension~2 theorems.
We study the lowest dimensional open case of the question whether every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^N$ is glicci, that is, whether every zero-scheme in $\mathbb{P}^3$ is glicci. We show that a set of $n \geq 56$ points in general position in $\PP^3$ admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in $\mathbb{P}^3$.
The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.
We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Miró-Roig.
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.
A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext^2(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dim_A(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by…
We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of codimension 3 we found several Gorenstein Betti sequences compatible with $H$ which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.
In this paper I investigate minimal surfaces of general type with p_g=5, q=0 for which the 1-canonical map is a birational morphism onto a surface in P^4 (so called canonical surfaces in P^4) via a structure theorem for the Hilbert resolutions of the canonical rings of the afore-mentioned surfaces, viewed as Gorenstein algebras of codimension 2 over the homogeneous coordinate ring of P^4. I discuss how the ring structure of such an algebra is encoded in its resolution. Among other things I show how this method can be applied to analyze the moduli space of canonical surfaces with p_g=5, q=0, K^…
We give some experimental data of Gorenstein liaison, working with points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$, to see how far the familiar situation of liaison, biliaison, and Rao modules of curves in ${\mathbb P}^3$ will extend to subvarieties of codimension 3 in higher ${\mathbb P}^4$.
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is in the Gorenstein liaison class of a complete intesection and we a…
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of th…
We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.
Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$. These are sheaves that correspond to the graded maximal Cohen--Macaulay modules on the homogeneous coordinate ring of $X$. Using known results on MCM modules, we are able to determine the Gorenstein biliaison classes of codimension two subschemes of certain varieties, including the nonsingular quadric surface in ${\mathbb P}^3$, and the cone over it in ${\ma…
We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being pe…
We extend the theory of generalized divisors so as to work on any scheme $X$ satisfying the condition $S_2$ of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof in a stronger form of the theorem of Gaeta, that standard determinantal schemes are in the Gorenstein biliaison class of a complete intersection. We also show, for schemes of codimension three in ${\mathbb P}^n$, that the relation of Gorenstein biliaison is equivalent to the relation of even strict Gorenstein liaison.
Let $X$ be a codimension 1 subvariety of dimension $>1$ of a variety of minimal degree $Y$. If $X$ is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then $X$ is Arithmetically Gorenstein and we characterise such subvarieties $X$ of $Y$ via apolarity as those whose apolar hypersurfaces are Fermat.
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…
A main ingredient for Kustin-Miller unprojection, as developed in (S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, math.AG/0011094), is the module Hom_R(I, \om_R), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of (F. Catanese et al., Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185-220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension…
In this survey article we want to discuss a way of constructing arithmetically Gorenstein varieties of high codimension. Consider kernel sheaves B_G of general, generically surjective morphisms G between decomposable bundles on P^n. The top-dimensional part of the zero-locus of a regular section of B_G is Gorenstein if B_G has odd rank. We show how to implement the construction method in the computer algebra system Singular and produce some examples of Gorenstein curves and threefolds in P^6. Finally, we investigate a class of sheaves B_G, where the degeneracy locus of G does not have the expe…
We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen-Macaulay modules, which we review in an Appendix.
We show that a Gorenstein subcanonical codimension 3 subscheme Z in X = P^N, N > 3, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately, and conversely. We extend this result to singular Z and all quasiprojective ambient schemes X under the necessary hypothesis that $Z$ is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of Buchsbaum-Eisenbud and says that Z is Pfaffian.…
In this paper we consider the problem of finding explicitly canonical ideals of one-dimensional Cohen-Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension two case, from a Hilbert-Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.
Given a zero-dimensional Gorenstein algebra $\mathbb{B}$ and two syzygies between two elements $f_1,f_2\in\mathbb{B}$, one constructs a double complex of $\mathbb{B}$-modules, ${\cal G}_\mathbb{B},$ called the small Gobelin. We describe an inductive procedure to construct the even and odd hyperhomologies of this complex. For high degrees, the difference $\dim \mathbb{H}_{j+2}({\cal G}_\mathbb{B}) - \dim\mathbb{H}_j({\cal G}_\mathbb{B})$ is constant, but possibly with a different value for even and odd degrees. We describe two flags of ideals in $\mathbb{B}$ which codify the above differences o…
This work is part of the Graded Ring Database project [GRDB], and is a sequel to [Altinok's 1998 PhD thesis] and [Altinok, Brown and Reid, Fano 3-folds, K3 surfaces and graded rings, in SISTAG (Singapore, 2001), Contemp. Math. 314, 2002, pp. 25-53]. We introduce a strategy based on Kustin-Miller unprojection that constructs many hundreds of Gorenstein codimension 4 ideals with 9x16 resolutions (that is, 9 equations and 16 first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds o…
