Homology Of Artinian Modules Over Commutative Noetherian Rings, 2011 University of Nebraska-Lincoln

#### Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

*Dissertations, Theses, and Student Research Papers in Mathematics*

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti ...

Characteristic Polynomial Of Arrangements And Multiarrangements, 2011 The University of Western Ontario

#### Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

*Electronic Thesis and Dissertation Repository*

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some ...

A Characterization Of Dynkin Elements, 2011 University of Massachusetts - Amherst

#### A Characterization Of Dynkin Elements, Pe Gunnells, E Sommers

*Paul Gunnells*

We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its $N$-region. In type $A_n$ this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells, which leads to some conjectures in representation theory.

Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, 2011 University of Massachusetts - Amherst

#### Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell

*Paul Gunnells*

In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms.

Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, 2011 University of Massachusetts - Amherst

#### Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, A Ash, Pe Gunnells, M Mcconnell

*Paul Gunnells*

In a previous paper [3] we computed cohomology groups H5(..0(N),C), where ..0(N) is a certain congruence subgroup of SL(4,Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.

Perverse Sheaves On Loop Grassmannians And Langlands Duality, 2011 University of Massachusetts - Amherst

#### Perverse Sheaves On Loop Grassmannians And Langlands Duality, I Mirkovic, K Vilonen

*Ivan Mirkovic*

No abstract provided.

Geometric Langlands Duality And Representations Of Algebraic Groups Over Commutative Rings, 2011 University of Massachusetts - Amherst

#### Geometric Langlands Duality And Representations Of Algebraic Groups Over Commutative Rings, I Mirkovic, K Vilonen

*Ivan Mirkovic*

No abstract provided.

On The Locus Of Hodge Classes, 2011 University of Massachusetts - Amherst

#### On The Locus Of Hodge Classes, E Cattani, P Deligne, A Kaplan

*Eduardo Cattani*

Let S be a nonsingular complex algebraic variety and V a polarized variation of Hodge structure of weight 2p with polarization form Q. Given an integer K, let S(K) be the space of pairs (s, u) with s ∈ S, u ∈ Vs integral of type (p, p), and Q(u, u) ≤ K. We show in Theorem 1.1 that S(K) is an algebraic variety, finite over S. When V is the local system H2p (Xs, Z)/torsion associated with a family of nonsingular projective varieties parametrized by S, the result implies that the locus where a given integral class ...

Frobenius Modules And Hodge Asymptotics, 2011 University of Massachusetts - Amherst

#### Frobenius Modules And Hodge Asymptotics, E Cattani, J Fernandez

*Eduardo Cattani*

We exhibit a direct correspondence between the potential defining the H1,1 small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules.

Complete Intersections In Toric Ideals, 2011 University of Massachusetts - Amherst

#### Complete Intersections In Toric Ideals, E Cattani, R Curran, A Dickenstein

*Eduardo Cattani*

We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals such that no binomial ideal contained in and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.

Asymptotic Hodge Theory And Quantum Products, 2011 University of Massachusetts - Amherst

#### Asymptotic Hodge Theory And Quantum Products, E Cattani, Javier Fernandez

*Eduardo Cattani*

Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold $X$ one may construct a polarized variation of Hodge structure over the complexified K\"ahler cone of $X$. In this paper we show that, in the case of fourfolds, there is a correspondence between ``quantum potentials'' and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths' trasversality property of a variation of Hodge structure.

Binomial Residues, 2011 University of Massachusetts - Amherst

#### Binomial Residues, E Cattani, A Dickenstein, B Sturmfels

*Eduardo Cattani*

A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A.

Mixed Lefschetz Theorems And Hodge-Riemann Bilinear Relations, 2011 University of Massachusetts - Amherst

#### Mixed Lefschetz Theorems And Hodge-Riemann Bilinear Relations, E Cattani

*Eduardo Cattani*

The Hard Lefschetz Theorem (HLT) and the Hodge–Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact Kähler manifold; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the f-vectors of convex polytopes. While the statements of these theorems depend on the choice of a Kähler class, or its analog, there is usually a cone of possible choices. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note, we present a unified approach ...

Residues And Resultants, 2011 University of Massachusetts - Amherst

#### Residues And Resultants, E Cattani, Alicia Dickenstein, Bernd Sturmfels

*Eduardo Cattani*

Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.

The A-Hypergeometric System Associated With A Monomial Curve, 2011 University of Massachusetts - Amherst

#### The A-Hypergeometric System Associated With A Monomial Curve, E Cattani, C D'Andrea, A Dickenstein

*Eduardo Cattani*

No abstract provided.

Counting Solutions To Binomial Complete Intersections, 2011 University of Massachusetts - Amherst

#### Counting Solutions To Binomial Complete Intersections, E Cattani, A Dickenstein

*Eduardo Cattani*

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

Rational Hypergeometric Functions, 2011 University of Massachusetts - Amherst

#### Rational Hypergeometric Functions, E Cattani, A Dickenstein, B Sturmfels

*Eduardo Cattani*

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator ...

Restriction Of A-Discriminants And Dual Defect Toric Varieties, 2011 University of Massachusetts - Amherst

#### Restriction Of A-Discriminants And Dual Defect Toric Varieties, R Curran, E Cattani

*Eduardo Cattani*

We study the A-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We give characterizations of dual defect toric varieties in terms of their Gale dual and classify dual defect toric varieties of codimension less than or equal to four.

Residues In Toric Varieties, 2011 University of Massachusetts - Amherst

#### Residues In Toric Varieties, E Cattani, D Cox, A Dickenstein

*Eduardo Cattani*

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees ...

Planar Configurations Of Lattice Vectors And Gkz-Rational Toric Fourfolds In P-6, 2011 University of Massachusetts - Amherst

#### Planar Configurations Of Lattice Vectors And Gkz-Rational Toric Fourfolds In P-6, E Cattani, A Dickenstein

*Eduardo Cattani*

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,)-equivalence and deduce that the only gkz-rational toric four-folds in 6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.