arXiv subjects

Emanuela Marangone

Publications and source records attributed to Emanuela Marangone.

8 recordsLinked to original sources

arXiv2026

Cohomology on the incidence correspondence and related questions

We study a variety of questions centered around the computation of cohomology of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it). Over a field of characteristic zero, this problem is resolved by the Borel-Weil-Bott theorem. In positive characteristic, we give recursive formulas for cohomology, generalizing work of Donkin and Liu in the case of the 3-dimensional flag variety. In characteristic 2, we provide non-recursive formulas describing the cohomology characters in terms of truncated Schur polynomials and Nim symmetric polynomials. The main technical ingredient in our work is the recursive description of the splitting type of vector bundles of principal parts on the projective line. We also discuss properties of the structure constants in the graded Han-Monsky representation ring, and explain how our cohomology calculation characterizes the Weak Lefschetz Property for Artinian monomial complete intersections.

math.AG↗

arXiv2026

Lefschetz properties for monomial complete intersections

We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of $\mathfrak{sl}_2$. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner--Kaid and Kustin--Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the $p$-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green--Han--Monsky ring.

math.AC↗

arXiv2026

Cohomology characters on the incidence correspondence

We investigate the cohomology of line bundles on the incidence correspondence, the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it. In characteristic zero, this cohomology is governed by the Borel-Weil-Bott theorem. In characteristic p>0, however, it becomes considerably subtler, and admits an equivalent reformulation in terms of cohomology tables for divided powers of the cotangent bundle on projective space. Our approach to the problem involves passing to infinitesimal thickenings of the incidence correspondence inside the ambient product of projective spaces. This leads to recursive formulas for the cohomology, generalizing earlier work of Donkin, of Liu, and of Gao-Raicu. We obtain generating functions for cohomology characters, expressed using truncated Schur polynomials and symmetric polynomials encoding the higher structure constants of the Verlinde algebras of SU(2) at levels (p-2) and (2p-2). Along the way, we exploit two important connections with multiplication in the graded Green-Han-Monsky representation ring: one relates this ring to cohomology, and another connects to the Verlinde algebras through the work of Coulembier-Etingof-Ostrik.

math.AG↗

arXiv2026

The non-Lefschetz locus of conics

A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\ell$ such that the multiplication map by $\ell:[A]_i\to [A]_{i+1}$ has maximum rank in every degree. The linear forms satisfying this property form a Zariski-open set; its complement is called the non-Lefschetz locus of $A$. In this paper, we investigate analogous questions for degree-two forms rather than lines. We prove that any complete intersection $A=k[x_1,x_2,x_3]/(f_1,f_2,f_3)$, with $\text{char } k=0$, has the Strong Lefschetz Property at range $2$, i.e. there exists a linear form $\ell\in [R]_1$, such that the multiplication map $\times \ell^2: [M]_i\to [M]_{i+2}$ has maximum rank in each degree. Then we focus on the forms of degree 2 such that $ \times C: [A]_i\to [A]_{i+2}$ fails to have maximum rank in some degree $i$. The main result shows that the non-Lefschetz locus of conics for a general complete intersection $A=k[x_1,x_2,x_3]/(f_1,f_2,f_3)$ has the expected codimension as a subscheme of $\mathbb{P}^5$. The hypothesis of generality is necessary. We include examples of monomial complete intersections in which the non-Lefschetz locus of conics has different codimension. To extend a similar result to the first cohomology modules of rank $2$ vector bundles over $\mathbb{P}^2$, we explore the connection between non-Lefschetz conics and jumping conics. The non-Lefschetz locus of conics is a subset of the jumping conics. Unlike the case of the lines, this can be proper when $\mathcal{E}$ is semistable with first Chern class even.

math.AC↗

arXiv2026

Geometrically vertex decomposable star configurations

The goal of this paper is to determine how the family of ideals of star configurations intersects with the class of geometrically vertex decomposable ideals. The main result of this paper shows that the answer is subtle since the geometrically vertex decomposability property of an ideal is not invariant under a linear change of variables, and thus the answer will depend upon the choice of the linear forms that define the ideal of the star configuration. We also show that the ideal of a star configuration is a Knutson ideal precisely when it is a geometrically vertex decomposable ideal.

math.AC↗

arXiv2026

Weighted Veronese Rings via Convex Semigroups

We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gröbner basis, graded Hilbert series and graded Betti numbers, the structure of their associated graded rings, and their Koszul property. We give examples in higher dimensions illustrating that the first and last properties may fail. Our approach leverages convex monomial ideals as introduced in Herzog-Qureshi-Saem(2019), which give rise to convex semigroups.

math.AC↗

arXiv2025

Computing the cohomology of line bundles on the incidence correspondence and related invariants

We describe the package "IncidenceCorrespondenceCohomology" for the computer algebra system Macaulay2. The main feature concerns the computation of characters and dimensions for the cohomology groups of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it). Additionally, the package provides tools for (1) computing the multiplication in the graded Han-Monsky representation ring, (2) determining the splitting type of vector bundles of principal parts on the projective line, and (3) testing the weak and strong Lefschetz properties for Artinian monomial complete intersections.

math.AG↗

arXiv2023

The non-Lefschetz locus of vector bundles of rank 2 over $\mathbb{P}^2$

A finite length graded $R$-module $M$ has the Weak Lefschetz Property if there is a linear element $\ell$ in $R$ such that the multiplication map $\times\ell: M_i\to M_{i+1}$ has maximal rank. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. In this paper we focus on the study of the non-Lefschetz locus for the first cohomology module $H_*^1(\mathbb{P}^2,\mathcal{E})$ of a locally free sheaf $\mathcal{E}$ of rank $2$ over $\mathbb{P}^2$. The main result is to show that this non-Lefschetz locus has the expected codimension under the assumption that $\mathcal{E}$ is general.

math.AG↗