arXiv subjects
Publications and source records attributed to Pedro Macias Marques.
A homogeneous bivariate $d$-form defines an $(i+1)$-rowed Toeplitz matrix for each $i$ between $0$ and $d$. We use Hodge theory and Schur polynomials to prove that if the $(i+1)$-rowed Toeplitz matrix of a form is totally nonnegative, then so is the $i$-rowed one. This fixes a gap in the main result of paper above.
In this paper, we compute all possible Jordan types of linear forms $\ell$ in any full Perazzo algebra $A$. In some cases we are also able to compute the corresponding Jordan degree type, which is a finer invariant.
We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that the subclass of pre-meadows with $\mathbf{a}$, coming from the lattice of open sets of a topological space $X$, and presheaves over $X$ are the same structure. Furthermore, we provide a construction that, given a sheaf of rings $\mathcal{F}$ on $X$ produces a common meadow as a disjoint union of elements of the form $\mathcal{F}(U)$ indexed over the open subsets of $X$. We also establish a correspondence between the process of going from a presheaf to a sheaf (called sheafification) and the process of going from a pre-meadow with $\mathbf{a}$ to a common meadow.
Let $K$ be a field, $I\subset R=K[x_1,\dots,x_n]$ and $J\subset T=K[y_1,\dots,y_m]$ be graded ideals. Set $S=R\otimes_KT$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions $\text{v}(I^k)$ and $\text{v}(J^k)$ is investigated. When $I$ and $J$ are monomial ideals, we describe $\text{v}(L^k)$, giving an explicit formula involving $\text{v}(I^k)$ and $\text{v}(J^k)$.
We consider Artinian algebras $A$ over a field $\mathsf{k}$, both graded and local algebras. The Lefschetz properties of graded Artinian algebras have been long studied, but more recently the Jordan type invariant of a pair $(\ell,A)$ where $\ell$ is an element of the maximal ideal of $A$, has been introduced. The Jordan type gives the sizes of the Jordan blocks for multiplication by $\ell$ on $A$, and it is a finer invariant than the pair $(\ell,A)$ being strong or weak Lefschetz. The Jordan degree type for a graded Artinian algebra adds to the Jordan type the initial degree of ``strings'' in the decomposition of $A$ as a $\mathsf{k}[\ell]$ module. We here give a brief survey of Jordan type for Artinian algebras, Jordan degree type for graded Artinian algebras, and related invariants for local Artinian algebras, with a focus on recent work and open problems.
A (standard graded) oriented Artinian Gorenstein algebra over the real numbers is uniquely determined by a real homogeneous polynomial called its Macaulay dual generator. We study the mixed Hodge-Riemann relations on oriented Artinian Gorenstein algebras for which we give a signature criterion on the higher mixed Hessian matrices of its Macaulay dual generator. Inspired by recent work of Brändén and Huh, we introduce a class of homogeneous polynomials in two variables called $i$-Lorentzian polynomials, and show that these are exactly the Macaulay dual generators of oriented Artinian Gorenstein algebras in codimension two satisfying mixed Hodge-Riemann relations up to degree $i$ on the positive orthant of linear forms. We further show that the set of $i$-Lorentzian polynomials of degree $d$ are in one-to-one correspondence with the set of totally nonnegative Toeplitz matrices of size depending on $i$ and $d$. A corollary is that all normally stable polynomials, i.e. polynomials whose normalized coefficients form a PF sequence, are $i$-Lorentzian. Another corollary is an analogue of Whitney's theorem for Toeplitz matrices, which appears to be new: the closure of the set of totally positive Toeplitz matrices, in the Euclidean space of all real matrices of a given size, is equal to the set of totally nonnegative Toeplitz matrices.
We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form.
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 $ω_{Q/J}$ into $Q/J$.
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.
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of the family Gor(H) of AG algebras: that by Jordan type satisfies the usual dominance property; the second filtration, by symmetric decomposition, satisfies a known semicontinuity property. Our examples are in codimension three -- the lowest codimension of such an example, as Gor(H) is irreducible in codimension two.
We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.
In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by change of coordinates together with multiplication by invertible matrices from both sides.
We study the symmetric subquotient decomposition of the associated graded algebras $A^*$ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra $A$. This decomposition arises from the stratification of $A^*$ by a sequence of ideals $A^*=C_A(0)\supset C_A(1)\supset\cdots$ whose successive quotients $Q(a)=C(a)/C(a+1)$ are reflexive $A^*$ modules. These were introduced by the first author, and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms. For us a Gorenstein sequence is an integer sequence $H$ occurring as the Hilbert function for an AG algebra $A$, that is not necessarily homogeneous. Such a Hilbert function $H(A)$ is the sum of symmetric non-negative sequences $H_A(a)=H(Q_A(a))$, each having center of symmetry $(j-a)/2$ where $j$ is the socle degree of $A$: we call these the symmetry conditions, and the decomposition $\mathcal{D}(A)=(H_A(0),H_A(1),\ldots)$ the symmetric decomposition of $H(A)$. We here study which sequences may occur as the summands $H_A(a)$: in particular we construct in a systematic way examples of AG algebras $A$ for which $H_A(a)$ can have interior zeroes, as $H_A(a)=(0,s,0,\ldots,0,s,0)$. We also study the symmetric decomposition sets $\mathcal{D}(A)$, and in particular determine which sequences $H_A(a)$ can be non-zero when the dual generator is linear in a subset of the variables. Several groups have studied "exotic summands" of the Macaulay dual generator $F$. Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no "exotic" summands. We apply this to Gorenstein algebras that are connected sums. We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.
Free extensions of commutative Artinian algebras were introduced by T. Harima and J. Watanabe. The Jordan type of a multiplication map $m$ by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for $m$. We show that a free extension of the Artinian algebra $A$ with fibre $B$ is a deformation of the usual tensor product. This has consequences for the generic Jordan types of $A,B$ and $C$, showing that the Jordan type of $C$ is at least that of the usual tensor product in the dominance order. We give applications to algebras of relative coinvariants of linear group actions on a polynomial ring.
T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras $C$ over a base $A$ with fibre $B$. The free extensions are deformations of the usual tensor product, when $C$ is also Gorenstein, so are $A$ and $B$, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator $F$ for $C$ as a homogeneous "polynomial" in $T$ and the dual variables for $B$, and given the dual generator for $B$, we give sufficient conditions on $F$ that ensure that $C$ is a free extension of $A={\sf k}[t]/(t^n)$ with fiber $B$. We give examples that explore the sharpness of the statements. We also consider a special set of coinvariant algebras $C$ which are free extensions of $A$, but which do not satisfy the sufficient conditions of our main result.
We introduce the ``skew apolarity lemma'' and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in {$\bigwedge^dV_{\mathbb{C}}$ with $d\leq 3$ and $\dim V_{\mathbb{C}} \leq 8$}. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.
Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fiber ring corresponds to a fiber bundle over a manifold. For rings, as with manifolds, it is a natural question to ask: to what extent do properties of the base and the fiber carry over to the extension? For example, if the base and fiber both satisfy a strong Lefschetz property, can we conclude the same for the extension? Or, more generally, can one determine the generic Jordan type for the extension given the generic Jordan types of the base and fiber? We address these questions using the relative coinvariant rings as prototypical models. We show that if $V$ is a vector space and if the subgroup $W$ of the general linear group Gl(V), is a non-modular finite reflection group and $K\subset W$ is a non parabolic reflection subgroup, then the relative coinvariant ring $R^K_W$ cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings $R^K_W$, some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples give rise to open questions concerning Lefschetz properties of certain algebras $A(m,n)$, related to combinatorial questions proposed and partially answered by G. Almqvist.
We generalise Fløystad'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.