Arithmetically Cohen–Macaulay Research Articles

Homogeneous ACM and Ulrich bundles on rational homogeneous spaces

Abstract In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.

Homogeneous ACM bundles on isotropic Grassmannians

Abstract In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles over isotropic Grassmannians of types 𝐵, 𝐶 and 𝐷 in terms of step matrices. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over these isotropic Grassmannians. So we classify all homogeneous ACM bundles over isotropic Grassmannians combining the results on usual Grassmannians by Costa and Miró-Roig. Moreover, if the irreducible initialized homogeneous ACM bundles correspond to some special highest weights, then they can be characterized by succinct forms.

The ACM property for unions of lines in P1×P2

This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in P3.

On the arithmetic Cohen–Macaulayness of varieties parameterized by Togliatti systems

Given any diagonal cyclic subgroup $\Lambda \subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ of degree $d$ which are invariants of $\Lambda$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case the projective toric variety $X_d$ parameterized by $(m_{1},\ldots, m_{r})$ is called a $GT$-variety with group $\Lambda$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.

Rank 3 ACM bundles on general hypersurfaces in [formula omitted

We prove that a general hypersurface in P5 of degree d≥3 does not support an indecomposable rank 3 arithmetically Cohen-Macaulay (ACM) bundle. This settles the base case of a generic version of a conjecture of Buchweitz, Greuel and Schreyer.

Special arrangements of lines: Codimension 2 ACM varieties in ℙ1 × ℙ1 × ℙ1

In this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen–Macaulay (ACM) varieties in [Formula: see text], called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view.

Monads on projective varieties

We generalise Fl\o{}ystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.

Multiprojective spaces and the arithmetically Cohen–Macaulay property

AbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially ( P 1 ) n (\mathbb P^1)^n . A combinatorial characterization, the ( ⋆ ) (\star ) -property, is known in P 1 × P 1 \mathbb P^1 \times \mathbb P^1 . We propose a combinatorial property, ( ⋆ s ) (\star _s) with 2 ≤ s ≤ n 2\leq s\leq n , that directly generalizes the ( ⋆ ) (\star ) -property to ( P 1 ) n (\mathbb P^1)^n for larger n n . We show that X X is ACM if and only if it satisfies the ( ⋆ n ) (\star _n) -property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.

On syzygies, degree, and geometric properties of projective schemes with property [formula omitted

Let X be a reduced, but not necessarily irreducible closed subscheme of codimension e in a projective space. One says that X satisfies property Nd,p(d≥2) if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree <d+i for 0≤i≤p (see [10] for details). Much attention has been paid to linear syzygies of quadratic schemes (d=2) and their geometric interpretations (cf. [1,9,15–17]). However, not very much is actually known about algebraic sets satisfying property Nd,p, d≥3. Assuming property Nd,e, we give a sharp upper bound deg⁡(X)≤(e+d−1d−1). It is natural to ask whether deg⁡(X)=(e+d−1d−1) implies that X is arithmetically Cohen–Macaulay (ACM) with a d-linear resolution. In case of d=3, by using the elimination mapping cone sequence and the generic initial ideal theory, we show that deg⁡(X)=(e+22) if and only if X is ACM with a 3-linear resolution. This is a generalization of the results of Eisenbud et al. (d=2) [9,10].We also give more general inequality concerning the length of the finite intersection of X with a linear space of not necessary complementary dimension in terms of graded Betti numbers. Concrete examples are given to explain our results.

Castelnuovo–Mumford regularity and arithmetic Cohen–Macaulayness of complete bipartite subspace arrangements

We give the Castelnuovo–Mumford regularity of arrangements of (n−2)-planes in Pn whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen–Macaulay.

Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙn. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

Rank 2 ACM bundles on complete intersection Calabi–Yau threefolds

The aim of this paper is to classify indecomposable rank two arithmetically Cohen–Macaulay (ACM) bundles on general complete intersection Calabi–Yau threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank two ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi–Yau threefolds as hypersurfaces). A strong indication leading to existence of bundles with \(c_1\,=\,2\), \(c_2\,=\,13\) on a quintic conjectured in Chiantini and Madonna (Le Matematiche 55:239–258, 2000), and Mohan Kumar and Rao (Cent Eur J Math 10(4):1380–1392, 2012) is found.

Frobenius Push-Forwards on Quadrics

We generalize and simplify Langer's results concerning Frobenius direct images of line bundles on quadrics, describing explicitly the decompositions of higher Frobenius push-forwards of arithmetically Cohen–Macaulay (ACM) bundles into indecomposables, with an additional emphasis on the case of characteristic two. These results are applied to check which Frobenius push-forwards of the structure sheaf are tilting.

ACM sets of points in multiprojective space

If $$\mathbb{X}$$ is a finite set of points in a multiprojective space $$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then $$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM

The minimal resolutions of double points in [formula omitted] with ACM support

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X , the support of Z , is arithmetically Cohen–Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X , for the bigraded Betti numbers of I Z , the defining ideal of Z . We then relate the total Betti numbers of I Z to the shifts in the graded resolution, thus answering a special case of a question of Römer.

Looking for minimal graded Betti numbers

We consider $O$-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes $X$ of codimension three in ${\pp}^n$. These are Hilbert functions $\varphi$ of Artinian algebras that are quotients of the coordinate ring of $X$ by a linear system of parameters. Using suitable decompositions of $\varphi$, we determine the minimal number of generators possible in some degree $c$ for the defining ideal of any such ACM scheme having the given $O$-sequence. We apply this result to construct Artinian Gorenstein $O$-sequences $\varphi$ of codimension $3$ such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function $\varphi$ admits more than one minimal element. Finally, for all $3$-codimensional complete intersection $O$-sequences we obtain conditions under which the corresponding poset of graded Betti sequences has more than one minimal element.

Arithmetically cohen-macaulay curves inP 4 of degree 4 and genus 0

We show that the arithmetically Cohen-Macaulay (ACM) curves of de- gree 4 and genus 0 in P4 form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the Hilbert scheme has dimension greater than 6. Moreover, we describe the structures of all ACM curves of Hilb 4m+1 (P 4 ).

On codimension-two subvarieties of hypersurfaces.

We show that for a smooth hypersurface $X\subset \bbP^n$ of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties $Y\subset X$ which are not an intersection $X\cap{S}$ for a codimension two subvariety $S\subset\bbP^n$. We also show there exist $Y\subset X$ as above for which the normal bundle sequence for the inclusion $Y\subset X\subset\bbP^n$ does not split.