arXiv subjects

Vasiliki Petrotou

Publications and source records attributed to Vasiliki Petrotou.

9 recordsLinked to original sources

arXiv2026

Frobenius identities for the volume map on Cohen--Macaulay rings

We study the volume map on Artinian quotients of Cohen-Macaulay algebras in characteristic $p$, and the interaction between it and the action of Frobenius on resolutions. This allows us to provide a general, conceptual way to understand Parseval-Rayleigh identities, curious inhomogeneous identities on the volume map which were developed for the proof of the Ohsugi-Hibi conjecture. This general perspective gives a new approach to generic Lefschetz theory. We use this perspective to do the following: we give sufficient conditions for anisotropy and the Hard Lefschetz property for generic Artinian reductions of graded Gorenstein rings; we study the codimension-$3$ Gorenstein quotient of a polynomial ring by the ideal generated by Pfaffians, proving a Parseval-Rayleigh identity and deriving anisotropy and Hard Lefschetz in characteristic $2$; we deduce the $g$-theorem for simplicial spheres and the Ohsugi-Hibi conjecture following previous work of Adiprasito, Papadakis, and Petrotou; and we provide further examples of Parseval-Rayleigh identities for Gorenstein rings.

math.AC↗

arXiv2025

Parseval-Rayleigh identities for homogeneous complete intersections

We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete intersections have the Strong Lefschetz Property over any field of characteristic 2.

math.AC↗

arXiv2025

Lattice polytopes and semigroup algebras: Generic Lefschetz properties and Parseval-Rayleigh identities

We study semigroup algebras associated to lattice polytopes. We begin by generalizing and refining work of Hochster, and describe the volume maps of these algebras, that is, their fundamental classes, in terms of Parseval-Rayleigh identities and differential equations, which we prove to be equivalent. We use these descriptions to establish strong Lefschetz properties. A consequence is the resolution of several conjectures concerning unimodality properties of the h*-polynomial of lattice polytopes.

math.CO↗

arXiv2024

The volume intrinsic to a commutative graded algebra

Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of these works, the behaviour of the canonical module under deformations in this moduli space. We demonstrate that even when there is no natural geometry around, we can give a viewpoint that behaves like it, effectively constructing geometry out of nothing, giving interpretation to intersection numbers without cycles. Moreover, we explore some properties of this normalization.

math.AC↗

arXiv2023

The 4-Intersection Unprojection Format

Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In the present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal which is contained in four codimension 3 complete intersection ideals and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space.

math.AG↗

arXiv2022

Beyond positivity in Ehrhart Theory

We study semigroup algebras arising from lattice polytopes, compute their volume polynomials (particularizing work of Hochster), and establish strong Lefschetz properties (generalizing work of the first three authors). This resolves several conjectures concerning unimodality properties of the $h^\ast$-polynomial of lattice polytopes arising within Ehrhart theory.

math.CO↗

arXiv2021

Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.

math.CO↗

arXiv2020

The characteristic 2 anisotropicity of simplicial spheres

Assume D is a simplicial sphere, and k_1 is a field. We say that D is generically anisotropic over k_1 if, for a certain purely transcendental field extension k of k_1, a certain Artinian reduction A of the Stanley-Reisner ring k[D] has the following property: All nonzero homogeneous elements u of A of degree less or equal to (dim D +1)/2 have nonzero square. We prove, using suitable differential operators, that, if the field k_1 has characteristic 2, then every simplicial sphere D is generically anisotropic over k_1. As an application, we give a second proof of a recent result of Adiprasito, known as McMullen's g-conjecture for simplicial spheres. We also prove that the simplicial spheres of dimension 1 are generically anisotropic over any field k_1.

math.AC↗

arXiv2020

Tom & Jerry triples with an application to Fano 3-folds

Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. The idea of unprojection is intended for serial use. Papadakis and Neves developed a theory of parallel unprojection. In the present work we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the pfaffians of a 5x5 skewsymmetric matrix, we use parallel unprojection of Kustin-Miller type in order to construct Gorenstein rings of codimension 6. We give two applications. These are two families of codimension 6 Fano 3-folds, in weighted projective space.

math.AG↗