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 $\mathbb{P}^3$.
We present a list of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.
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.
We characterize the postulation character of arithmetically Gorenstein curves in ${\mathbb P}^4$. We give conditions under which the curve can be realized in the form $mH-K$ on some ACM surface. Finally, we strengthen a theorem of Watanabe by showing that any general arithmetically Gorenstein curve in ${\mathbb P}^4$ can be obtained from a line by a series of ascending complete-intersection biliaisons.
In this paper we compute the Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes that are twisted anti-canonical divisors on arithmetically Cohen-Macaulay schemes. We also prove some folklore results characterizing the Hilbert functions of irreducible standard determinantal schemes, and we use them to produce a new class of functions that occur as Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes in any codimension.
We characterize the minimal free resolution of zero-dimensional subschemes in the plane with non connected character. This is then used to slightly generalize a result of Sauer about the smoothability of a.C.M. space curves. Some complements are also given.
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration $G$ of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the Weak Lefschetz Property. Furthermore, we s…
A central problem in liaison theory is to decide whether every arithmetically Cohen-Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can be indeed achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen-Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of …
A locally free resolution of a subscheme is by definition an exact sequence consisting of locally free sheaves (except the ideal sheaf) which has uniqueness properties like a free resolution. The purpose of this paper is to characterize certain locally Cohen-Macaulay subschemes by means of locally free resolutions. First we achieve this for arithmetically Buchsbaum subschemes. This leads to the notion of an $Ω$-resolution and extends a result of Chang. Second we characterize quasi-Buchsbaum subschemes by means of weak $Ω$-resolutions. Finally, we describe the weak $Ω$-resolutions which belong …
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…
Let k be an algebraically closed field of characteristic 0. The question of irreducibility of the punctual Hilbert scheme Hilb_d P^n and its Gorenstein locus for various d was studied in [CEVV8, CN9, CN10, CN11]. In this short paper we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1,5,5,1) are smoothable i.e. lie in the closure of the locus of smooth subschemes. Among the Gorenstein algebras of length 12 the smoothability of algebras having such Hilbert function seems to be the most inapproachable using non-direct tools e.g. structural theorems.
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.
Let X be a nonsingular arithmetically Cohen-Macaulay projective scheme, Z a nonsingular subscheme of X. Let π: Y --> X be the blowup of X along the ideal sheaf of Z, E_0 the pull-back of a general hyperplane in X and E the exceptional divisor. In this paper, we study projective embeddings of Y given by the divisor tE_0 - eE. We give explicit values of d and δsuch that for all e > 0 and t > ed + δ, these embeddings is projectively normal and arithmetically Cohen-Macaulay. We also study the regularity of the ideal sheaf and syzygies of these embeddings. When X is a surface and Z is a 0-dimension…
In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results.
Let k be an algebraically closed field and let HaG(d) be the open locus inside H(d) (the Hilbert scheme of 0-dimensional length d subschemes of the projective (d-2)-space over k) corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of HaG(6). In order to achieve these results we also classify all Artinian, Gorenstein, not necessarily graded, k-algebras up to degree 6. Moreover we describe the loci in HaG(6) obtained via some geometric construction. Finally we prove the obstructedness of some families of points in HaG(d) for each …
A Pfaff field on a projective space is a map from the sheaf of differential s-forms, for a certain s, to an invertible sheaf. The interesting ones are those arising from a Pfaff system, as they give rise to a distribution away from their singular locus. A subscheme of the projective space is said to be invariant under the Pfaff field, if the latter induces a Pfaff field on the subscheme. We give bounds for the Castelnuovo-Mumford regularity of invariant complete intersection subschemes (more generally, arithmetically Cohen-Macaulay subschemes) of dimension s, depending on how singular these sc…
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.
We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $…
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…
Let $k$ be an algebraically closed field of characteristic 0 and let $H_G(d,N)$ be the open locus of the Hilbert scheme $H(d,N)$ corresponding to Gorenstein subschemes of degree $d$ in the projective N-space. We proved in a previous paper that $H_G(d,N)$ is irreducible for $d\le9$ and $N\ge1$. In the present paper we prove that also $H_G(10,N)$ is irreducible for each $N\ge1$, giving also a complete description of its singular locus.
For an arithmetically Cohen--Macaulay subscheme of projective space, there is a well-known bound for the highest degree of a minimal generator for the defining ideal of the subscheme, in terms of the Hilbert function. We prove a natural extension of this bound for arbitrary locally Cohen--Macaulay subschemes. We then specialize to curves in $\pthree$, and show that the curves whose defining ideals have generators of maximal degree satisfy an interesting cohomological property. The even liaison classes possessing such curves are characterized, and we show that within an even liaison class, all …
In this paper we study the problem of describing the integral subschemes within a fixed even linkage class $Ł$ of subschemes in $\Pn$ of codimension two. In the case that $Ł$ is not the class of arithmetically Cohen-Macaulay subschemes, we associate to any $X \in Ł$ two invariants $θ_X$ and $η_X$. When taken with the height $h_X$, each of these invariants determines the location of $X$ in $Ł$, thought of as a poset under domination. In terms of these invariants, necessary conditions are given for integral subschemes. The necessary conditions are almost sufficient in the sense that if a subsche…
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…
Generalizing an old result proved by P. Rao [see MR 83i:14025] for arithmetically Cohen-Macaulay, self-linked subschemes of codimension 2 in the projective n-space P, we give a characterization of self-linked pure subschemes of codimension 2 in P satisfying a necessary parity condition, in characteristic different from 2. We make use of the theory of quadratic sheaves described in a previous paper of the authors [see MR 99h:14044].
Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are union of complete intersections we give a structure theorem for arithmetically Cohen-Macaulay union of two complete intersections of codimension $2.$ We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.
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.
It is well known that for a subscheme $V$ in ${\mathbb P}^{n}$ of codimension two, the conditions (1) $V$ is ACM, and (2) $V$ is "licci" (i.e. $V$ is in the liaison class of a complete intersection) are equivalent. In higher codimension, (2) implies (1) but the converse is false. For Gorenstein liaison the analog of (2) is: $(2')$ $V$ is "glicci" (i.e. $V$ is in the Gorenstein liaison class of a complete intersection). It remains true that $(2')$ implies (1) but now the converse is an important open question. Previously the authors described a mechanism for lifting monomial ideals to reduced u…
A good way of parametrizing 0-dimensional schemes in an affine space $\mathbb{A}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $μ$, they provide an open covering of the Hilbert scheme ${\rm Hilb}^μ(\mathbb{A}^n_K)$ and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent 0-dimensional $K$-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The p…
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 subscheme $X\subset \Bbb P^{n+3}$ of codimension $3$ is {\em Pfaffian} if it is the degeneracy locus of a skew-symmetric map $f:\cal{E}\spcheck(-t) @>>> \cal{E}$ with $\cal{E}$ a locally free sheaf of odd rank on $\Bbb P^{n+3}$. It is shown that a codimension $3$ subscheme $X\subset\Bbb P^{n+3}$ is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e.\ $ω_X\cong\cal O_X(l)$ for some integer $l$), and the following parity condition holds: if $n\equiv 0\pmod{4}$ and $l$ is even, then $χ(\cal O_X (l/2))$ is also even. The paper includes a modern version of the Horrocks correspond…
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.…
Given a set of distinct points $X=\{P_1, \dots,P_r\} $ in $ \mathbb P^n_{\res}, $ in this paper we characterize being $X$ arithmetically Gorenstein through the ``special" structure of the inverse system of the defining ideal $I(X) \subseteq R= \res[X_0, \dots, X_n].$ We describe the corresponding Zariski locally closed subset of $\mathbb (\mathbb P_{\res}^{n})^r$ parametrized by the coordinates of the points. Several examples are given to show the effectiveness of the results. As a consequence of the main result we characterize the Artinian Gorenstein rings which are the Artinian reduction of …
Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we characterize geometrically its singularities for $d\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in $\Hilb_{d}^{G}(\p{N})$.
Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (I,f) consisting of a torsion-free, rank-1 sheaf I on C and a map of vector spaces f to the space of global sections of I. If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C(t) degenerating to C, and each family of linear systems (L(t),f(t)) along C(t), with L(t) invertible, degenerating to (I,f), the corresponding Weierstrass d…
The postulation of Arithmetically Cohen-Macaulay (ACM) subschemes of the projective space ${\mathbb P}^N_k$ is well-known in the case of codimension 2. There are many different ways of recording this numerical information : numerical character of Gruson/Peskine, $h$-vector, postulation character of Martin-Deschamps/Perrin... The first aim of this paper is to show the equivalence between these notions. The second, and most important aim, is to study the postulation of codimension 3 ACM subschemes of ${\mathbb P}^N$. We use a result of Macaulay which describes all the Hilbert functions of the qu…
In this paper the author provides a generalization of classical linkage, i.e. linkage by a complete intersection of dim. 0 or 1 on arithmetically Cohen-Macaulay schemes of any dimension. Namely she looks at residuals in the scheme theoretic intersection of an aCM scheme of dim. r (resp. r+1) with r hypersurfaces of degree a_1,...,a_r. When the aCM scheme is singular a complete intersection on it may not be Gorenstein, in this case classical linkage, even if suitably generalized, does not apply. The main purpose of this paper is to prove some results related to the invariance of the deficiency …
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…
Given a 0-dimensional scheme $\mathbb{X}$ in the projective $n$-space $\mathbb{P}^n_K$ over a field $K$, we are interested in studying the Kähler different of $\mathbb{X}$ and its applications. Using the Kähler different, we characterize the generic position and Cayley-Bacharach properties of $\mathbb{X}$ in several certain cases. When $\mathbb{X}$ is in generic position, we prove a generalized version of the Apéry-Gorenstein-Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the Kähler different and the Cayley-Bacharac…
The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group $G$. To achieve this, we introduce the moduli functor $\mathsf{M}^G_h$ of $G$-marked Gorenstein canonical models with Hilbert polynomial $h$, and prove the existence of $\mathfrak{M}_h[G]$, the coarse moduli scheme for $\mathsf{M}^G_h$. Then we show that $\mathfrak{M}_h[G]$ has a proper and finite morphism onto $\mathfrak{M}_h$ so that its image $\mathfrak{M}_h(G)$ is a closed subscheme. In the end we obtain the canonical r…
The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf …
If $V$ is an equidimensional codimension $c$ subscheme of an $n$-dimensional projective space, and $V$ is linked to $V'$ by a complete intersection $X$, then we say that $V$ is {\em minimally linked} to $V'$ if $X$ is a codimension $c$ complete intersection of smallest degree containing $V$. Gaeta showed that if $V$ is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of minimal links beginning with $V$ and arriving at a complete intersection. We extend this work in the following ways: 1) In the codimension 2 non-ACM case, we show that for any…
In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for constructing flat families and analysis of finite Gorenstein algebras and expose their efficiency by proving smoothability of certain families of algebras. Much of the existing theory and folklore is reviewed, providing a micro-encyclopaedic reference.
Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers d…
We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X \subset {\mathbb P}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $\tilde {X} \subset {\mathbb P}^{r + 1}$ from an appropriate point $p \in {\mathbb P}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $X$ by means of the projection $\tilde {X} \to X$. If $X$ is not arit…
A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph …
We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of the coordinate ring of the arrangement. We also prove that every (d-1)-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth $\ge 3$. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang.
Let $ϕ$ be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $ϕ$ has the expected codimension. We call $B_ϕ = \ker ϕ$ a (first) Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing $ϕ$ as a matrix (after choosing bases), we say that $X$ is good if one can delete a generalized row from $ϕ$ and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown …
The authors construct the global Macaulay inverse system for a zero-dimensional subscheme Z of projective n-space P^n, from the local inverse systems of the irreducible components of Z. They show that when Z is locally Gorenstein a generic homogeneous form F of degree d apolar to Z determines Z when d is larger than an invariant b(Z). They also show that a natural upper bound for the Hiilbert function of Gorenstein Artin quotient of the coordinate ring is achieved for large socle degree. They show the uniqueness of generalized additive decompositions of a homogeneous form F into powers of line…
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano Kähler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good …
Let X and Y be smooth varieties of dimensions n-1 and n over an arbitrary algebraically closed field, f:X-> Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, at every point of X, the Jacobian has rank at least n-2. For r at least 1, consider the subscheme N_r of Y defined by the (r-1)st Fitting ideal of the O_Y-module f_*O_X, and set M_r:=f^{-1}N_r. In this setting --- in fact, in a more general setting --- we prove the following statements, which show that M_r and N_r behave like reasonable schemes of source and target r-fold points of f. Each component…
