arXiv subjects

Claudiu Raicu

Publications and source records attributed to Claudiu Raicu.

At least 19 recordsLinked to original sources

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

Powers of binary forms and derived Hermite reciprocity

For $a,b \ge 1$, Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X, showing that it is generated in degree b+1 and that it has a linear minimal free resolution. We do this by generalizing results of Abdesselam and Chipalkatti on an analogue of the Foulkes--Howe map and by establishing a derived analogue of the classical Hermite reciprocity theorem for complexes of ${\rm SL}_2$-representations. In our investigation, we are led to the ideal generated by the subrepresentation ${\rm Sym}^{ab}({\Bbb C}^2) \subset {\rm Sym}^a({\rm Sym}^b {\Bbb C}^2)$. We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.

math.AC↗

arXiv2026

Diagonal F-thresholds for determinants and Pfaffians

We compute the diagonal F-thresholds of determinantal hypersurfaces arising from a generic matrix and from a generic symmetric matrix, as well as of the Pfaffian hypersurface arising from a generic skew-symmetric matrix of even size. The main ingredient is a cohomology vanishing theorem for certain line bundles on flag varieties in characteristic $p$. In the cases of the generic matrix and the generic skew-symmetric matrix, we show that the diagonal F-threshold attains its minimal possible value, namely the negative of the a-invariant. The symmetric case is more subtle and relies in addition on a polynomiality result for representations afforded by cohomology, building on work of the second author with VandeBogert.

math.AC↗

arXiv2025

The effective Chen ranks conjecture

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. We give applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of every arrangement whose projective resonance is reduced. Finally, we formulate a sharp generic vanishing conjecture for Koszul modules and present a parallel between this statement and the Prym--Green Conjecture on syzygies of general Prym canonical curves.

math.AG↗

arXiv2025

Stable sheaf cohomology and Koszul--Ringel duality

We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul--Ringel duality on the category of strict polynomial functors as described in the work of Cha\l upnik, Krause, and Touz\'e. Combining this with recent results of Maliakas--Stergiopoulou we confirm a conjectured periodicity statement for stable cohomology. In a different direction, we find a remarkable invariance property for $\Ext$ groups between Schur functors associated to hook partitions, and compute all such extension groups over a field of arbitrary characteristic. We show that this is further equivalent to the calculation of $\Ext$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker's recursive description of $\Ext$ groups for $\SL_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero.

math.RT↗

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↗

arXiv2024

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↗

arXiv2024

Reduced resonance schemes and Chen ranks

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

math.AG↗

arXiv2024

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle Δ\rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $Δ$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

math.AC↗

arXiv2023

Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.

math.AG↗

arXiv2023

Some questions arising from the study of cohomology on flag varieties

A fundamental problem at the confluence of algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties over a field of characteristic p. When p=0, the solution is given by the celebrated Borel-Weil-Bott Theorem, while for p>0 the problem is widely open. In this note we describe a collection of open questions that arise from the study of particular cases of the general theory, focusing on their combinatorial and commutative algebra aspects.

math.AG↗

arXiv2023

Stable sheaf cohomology on flag varieties

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.

math.AG↗

arXiv2022

Cohomology of line bundles on the incidence correspondence

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X in P(V) x P(V*), parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel-Weil-Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo-Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

math.AG↗

arXiv2021

Borel-Moore homology of determinantal varieties

We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the Čech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore homology of their closures, based on Saito's theory of mixed Hodge modules.

math.AG↗

arXiv2021

Local Euler obstructions for determinantal varieties

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaffney--Grulha--Ruas, for skew-symmetric matrices by Promtapan and Rimányi, and for all cases by Zhang.

math.AG↗

arXiv2021

On some modules supported in the Chow variety

The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups.

math.AC↗

arXiv2021

Bi-graded Koszul modules, K3 carpets, and Green's conjecture

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality.

math.AC↗