arXiv subjects

Uwe Nagel

Publications and source records attributed to Uwe Nagel.

At least 91 records · Page 5Linked to original sources

arXiv2007

Glicci simplicial complexes

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and on Stanley decompositions.

math.AC↗

arXiv2007

Specializations of Ferrers ideals

We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties.

math.AC↗

arXiv2007

Algebraic shifting and exterior and symmetric algebra methods

We establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shift is the best among the exterior shifting operations in the sense that it increases the number of facets the least. Methods of proof include Gröbner basis theory over the exterior algebra, Cartan homology, degree functions, and Alexander duality.

math.AC↗

arXiv2007

An improved Multiplicity Conjecture for codimension three Gorenstein algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.

math.AC↗

arXiv2006

Liaison addition and the structure of a Gorenstein liaison class

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.

math.AG↗

arXiv2006

Monomial and toric ideals associated to Ferrers graphs

Each partition $λ= (λ_1, λ_2, ..., λ_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

math.AC↗

arXiv2006

Bounds for the Castelnuovo-Mumford regularity of modules

We establish bounds for the Castelnuovo-Mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations. We extend to modules (and improve) the known bounds for homogenous ideal in a polynomial ring established by Galligo, Guisti, Caviglia and Sbarra.

math.AC↗

arXiv2005

Empty simplices of polytopes and graded Betti numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.

math.AC↗

arXiv2005

Families of space curves with large cohomology

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show that, if the cohomology is ``not too small,'' then they can be parameterized by the union of two generically smooth irreducible families; one of them corresponds to the subextremal curves. For curves of negative genus, the general curve of each of these families is also a smooth point of the support of an irreducible component of the Hilbert scheme. The two components have the same (large) dimension and meet in a subscheme of codimension one.

math.AG↗

arXiv2005

The equations of space curves on a quadric

The homogenous ideals of curves in a double plane have been studied by Chiarli, Greco, Nagel. Completing this work we describe the equations of any curve that is contained in some quadric. As a consequence, we classify the Hartshorne-Rao modules of such curves.

math.AG↗

arXiv2005

On the componentwise linearity and the minimal free resolution of a tetrahedral curve

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this work by showing that they are "almost always" componentwise linear, i.e. their homogeneous ideals have the property that for any d, the degree d component of the ideal generates a new ideal whose minimal free resolution is linear. The one type of exception is clearly spelled out and studied as well. The main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves, and the connection to the tetrahedral curves that are minimal in their even liaison classes. With this preparation, we also describe the minimal free resolution of a tetrahedral curve, and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution. Finally, we begin the study of the generic initial ideal (gin) of a tetrahedral curve. We produce the gin for arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class.

math.AC↗

arXiv2005

Minimal free resolutions of projective subschemes of small degree

We discuss the minimal free resolution of an irreducible projective subscheme X. If X is also reduced, we focus on the case when its degree equals two plus the codimension. The set of all possible graded Betti numbers is described if the codimension is two. In general, there are only partial results. Then we discuss Cohen-Macaulay structures on a linear subspace. For such curves of degree two, the classification has been obtained by Notari, Spreafico, and the author. In general, the classification problem seems rather difficult. Vatne has shown that such a classification of structures whose degree is at most three, is equivalent to Hartshorne's conjecture on smooth varieties of codimension two.

math.AG↗

arXiv2005

Extensions of the Multiplicity Conjecture

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-space, and we propose an upper bound for finitely generated graded torsion modules. We establish this bound for torsion modules whose codimension is at most two.

math.AC↗

arXiv2004

The Multiplicity Conjecture in low codimensions

We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we prove stronger bounds than the conjectured ones allowing us to characterize the extremal cases. This may be seen as a converse to the multiplicity formula of Huneke and Miller that inspired the conjectural bounds.

math.AC↗

arXiv2004

Liaison classes of modules

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 perfect, Cohen-Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module liaison classes. Furthermore, we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one.

math.AC↗

arXiv2004

Tetrahedral Curves

A tetrahedral curve is a space curve whose defining ideal is an intersection of powers of monomial prime ideals of height two. It is supported on a tetrahedral configuration of lines. Schwartau described when certain such curves are ACM, namely he restricted to curves supported on a certain four of the six lines. We consider the general situation. We first show that starting with an arbitrary tetrahedral curve, there is a particular reduction that produces a smaller tetrahedral curve and preserves the even liaison class. We call the curves that are minimal with respect to this reduction S-minimal curves. Given a tetrahedral curve, we describe a simple algorithm (involving only integers) that computes the S-minimal curve of the corresponding even liaison class; in the process it determines if the original curve is arithmetically Cohen-Macaulay or not. We also describe the minimal free resolution of an S-minimal curve, using the theory of cellular resolutions. This resolution is always linear. This result allows us to classify the arithmetically Buchsbaum, non-ACM tetrahedral curves. More importantly, it allows us to conclude that an S-minimal curve is minimal in its even liaison class; that is, the whole even liaison class can be built up from the S-minimal curve. Finally, we show that there is a large set of S-minimal curves such that each curve corresponds to a smooth point of a component of the Hilbert scheme and that this component has the expected dimension.

math.AC↗

arXiv2004

Extended degree functions and monomial modules

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.

math.AC↗

arXiv2004

Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra $R/\fc$, whose Hilbert function is maximal among such quotients with the given socle degrees. For us $\fc$ is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fröberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection. When the algebra is Gorenstein of even socle degree we give the precise resolution. When it is of odd socle degree we give good bounds on the graded Betti numbers. We also relate this case to the Minimal Resolution Conjecture of Mustata for points on a projective variety. Finding the graded Betti numbers is essentially equivalent to determining to what extent there can be redundant summands (i.e. ``ghost terms'') in the minimal free resolution, i.e. when copies of the same $R(-t)$ can occur in two consecutive free modules. This is easy to arrange using Koszul syzygies; we show that it can also occur in more surprising situations that are not Koszul. Using the equivalence to the Fröberg Conjecture, we show that in a polynomial ring where that conjecture holds (e.g. in three variables), the possible non-Koszul ghost terms are extremely limited. Finally, we use the connection to the Fröberg Conjecture, as well as the calculation of the minimal free resolution for relatively compressed Gorenstein algebras, to find the minimal free resolution of general Artinian almost complete intersections in many new cases. This greatly extends previous work of the first two authors.

math.AC↗