Artinian algebras and Jordan type

The Jordan type of an element $\ell$ of the maximal ideal of an Artinian k-algebra A acting on an A-module M of k-dimension n, is the partition of n given by the Jordan block decomposition of the multiplication map $m_\ell$ on M. In general the Jordan type has more information than whether the pair $(\ell,M)$ is strong or weak Lefschetz. We develop basic properties of the Jordan type and their loci for modules over graded or local Artinian algebras. We as well study the relation of generic Jordan type of $A$ to the Hilbert function of $A$. We introduce and study a finer invariant, the Jordan degree type. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.Comment: 53 pages. Added results, examples for Jordan degree type (Section 2.4) and Jordan type and initial ideal (Section 2.5

Stability of syzygy bundles

We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with a different choice of families of monomials [Coa09]. For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.Comment: 16 pages, to appear in the Proceedings of the American Mathematical Societ

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur

Artinian Gorenstein algebras of embedding dimension four and socle degree three

We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with $\mathsf{k}$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the $Q$-module $Q/I$ can be completely described in terms of a graded minimal free resolution of the $Q$-module $Q/J$ and a homogeneous embedding of a shift of the canonical module $\omega_{Q/J}$ into $Q/J$.Comment: 42 page

Splitting criteria for vector bundles on the symplectic isotropic Grassmannian

We extend a theorem of Ottaviani on cohomological splitting criterion for vector bundles over the Grassmannian to the case of the symplectic isotropic Grassmanian. We find necessary and sufficient conditions for the case of the Grassmanian of symplectic isotropic lines. For the general case the generalization of Ottaviani’s conditions are sufficient for vector bundles over the symplectic isotropic Grassmannian. By a calculation in the program LiE, we find that Ottaviani’s conditions are necessary for Lagrangian Grassmannian of isotropic k-planes for k ≤ 6, but they fail to be necessary for the case of the Lagrangian Grassmannian of isotropic 7-planes. Finally, we find a related set of necessary and sufficient splitting criteria for the Lagrangian Grassmannian

Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Artinian Gorenstein Algebras in Codimension Two

A homogeneous bivariate polynomial $F=F(X,Y)$ of degree $d$ with coefficients in $\mathbb{R}$ is the Macaulay dual generator of a codimension two standard graded oriented Artinian Gorenstein $\mathbb{R}$-algebra $A_F$ of socle degree $d$. We show that the total positivity of a certain Toeplitz matrix determined by the coefficients of $F$ is equivalent to a certain mixed Hodge-Riemann relation on the algebra $A_F$; polynomials satisfying these conditions are called higher Lorentzian polynomials. In degree $i=1$, our conditions amount to nonnegative ultra log concavity with no internal zeros, and we recover results of Br\"and\'en-Huh in $n=2$ variables. A corollary of our results is that the closure of the set of totally positive Toeplitz matrices is the set of totally nonnegative Toeplitz matrices, which seems to be new.Comment: Rewritten with new results, authorship change, comments still welcome!

LORENTZIAN POLYNOMIALS, HIGHER HESSIANS, AND THE HODGE-RIEMANN PROPERTY FOR CODIMENSION TWO GRADED ARTINIAN GORENSTEIN ALGEBRAS

We study the Hodge-Riemann property (HRP) for graded Artinian Gorenstein (AG) algebras. We classify AG algebras in codimension two that have HRP in terms of higher Hessian matrices and positivity of Schur functions associated to certain rectangular partitions. In this paper we introduce the Hodge Riemann property (HRP) on an arbitrary graded oriented Artinian Gorenstein (AG) algebra defined over R, and we give a criterion on the higher Hessian matrix of its Macaulay dual generator (Theorem 3.1). AG algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of complex manifolds, and the HRP is analogous to the Hodge-Riemann relations (HRR) satisfied by cohomology rings of complex Kähler manifolds. Higher Hessians were introduced by the fourth author [10] to study the strong Lefschetz property (SLP) of an AG algebra defined over an arbitrary field of characteristic zero (see also [4]); over the real numbers, HRP implies SLP (Lemma 2.3)