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IncidenceCorrespondenceCohomology -- Cohomology on the incidence correspondence, bundle of principal parts, Han-Monsky multiplication, and Lefschetz properties

Description

This package computes the sheaf cohomology of line bundles on the incidence correspondence, the torus equivariant splitting type of the bundle of principal parts for $\mathcal{O}(d)$ on $\mathbb{P}^1$, the multiplication in the graded Han-Monsky representation ring, and checks the Weak Lefschetz property.

This Macaulay2 package implements algorithms based on the theoretical results from

  • Annet Kyomuhangi, Emanuela Marangone, Claudiu Raicu, and Ethan Reed, Cohomology on the incidence correspondence and related questions, https://arxiv.org/abs/2411.13450 arXiv, 2024
  • Annet Kyomuhangi, Emanuela Marangone, Claudiu Raicu, and Ethan Reed, Computing the cohomology of line bundles on the incidence correspondence and related invariants,arXiv, 2025

This package also checks Strong Lefschetz Property in positive characteristic based on the classification from

  • Lisa Nicklasson, The strong Lefschetz property of monomial complete intersections in two variables, Collectanea Mathematica, 2018

Notation:

We indicate $\Omega$ the tangent bundle on $\mathbb{P}^{n-1}$, and with $\mathcal{R}=\Omega(1)$ the universal rank (n-1) subsheaf. $D^d \mathcal{R}$ denotes the d-th divided power of $\mathcal{R}$.

We indicate the incidence correspondence with $X$. Each line bundle on $X$ is the restriction of a line bundle on $\mathbb{P}\times\mathbb{P}^\vee$: we denote with $\mathcal{O}_X(a,b)$ the restriction to $X$ of the line bundle $\mathcal{O}(a)\times\mathcal{O}(b)$.

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Authors

Version

This documentation describes version 0.1 of IncidenceCorrespondenceCohomology.

Citation

If you have used this package in your research, please cite it as follows:

@misc{IncidenceCorrespondenceCohomologySource,
  title = {{IncidenceCorrespondenceCohomology: A \emph{Macaulay2} package. Version~0.1}},
  author = {Annet Kyomuhangi and Emanuela Marangone and Claudiu Raicu and Ethan Reed},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • hanMonsky -- Computes the multiplication in the graded Han-Monsky representation ring
    • hasSLP -- Checks whether a monomial complete intersection over a field of characteristic p has the Strong Lefschetz Property (SLP)
    • hasWLP -- Checks whether a graded Artinian algebra has the Weak Lefschetz Property (WLP)
    • incidenceCohomology -- Computes dimension or character of sheaf cohomology of line bundles on the incidence correspondence
    • monomialCIsWithoutWLP -- Lists all the monomial complete intersections of socle s and height n that fail the Weak Lefschetz Property
    • nimDividedCohomology -- Computes the dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space, in characteristic 2
    • recursiveDividedCohomology -- Computes dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space
    • splittingFdr -- Computes the torus equivariant splitting type of the bundles $\mathcal{F}^d_r$ introduced in [KMRR,24]
    • splittingPrincipalParts -- Computes the torus equivariant splitting type of the bundle of principal parts on $\mathbb{P}^1$
  • Methods
    • hanMonsky(ZZ,List) -- see hanMonsky -- Computes the multiplication in the graded Han-Monsky representation ring
    • hasSLP(ZZ,List) -- see hasSLP -- Checks whether a monomial complete intersection over a field of characteristic p has the Strong Lefschetz Property (SLP)
    • hasWLP(PolynomialRing,Ideal) -- see hasWLP -- Checks whether a graded Artinian algebra has the Weak Lefschetz Property (WLP)
    • hasWLP(ZZ,List) -- see hasWLP -- Checks whether a graded Artinian algebra has the Weak Lefschetz Property (WLP)
    • incidenceCohomology(List) -- see incidenceCohomology -- Computes dimension or character of sheaf cohomology of line bundles on the incidence correspondence
    • incidenceCohomology(List,PolynomialRing) -- see incidenceCohomology -- Computes dimension or character of sheaf cohomology of line bundles on the incidence correspondence
    • monomialCIsWithoutWLP(ZZ,ZZ,ZZ) -- see monomialCIsWithoutWLP -- Lists all the monomial complete intersections of socle s and height n that fail the Weak Lefschetz Property
    • nimDividedCohomology(ZZ,ZZ,ZZ,PolynomialRing) -- see nimDividedCohomology -- Computes the dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space, in characteristic 2
    • nimDividedCohomology(ZZ,ZZ,ZZ,ZZ) -- see nimDividedCohomology -- Computes the dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space, in characteristic 2
    • recursiveDividedCohomology(List) -- see recursiveDividedCohomology -- Computes dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space
    • recursiveDividedCohomology(List,PolynomialRing) -- see recursiveDividedCohomology -- Computes dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space
    • splittingFdr(ZZ,ZZ,ZZ) -- see splittingFdr -- Computes the torus equivariant splitting type of the bundles $\mathcal{F}^d_r$ introduced in [KMRR,24]
    • splittingPrincipalParts(ZZ,ZZ,ZZ) -- see splittingPrincipalParts -- Computes the torus equivariant splitting type of the bundle of principal parts on $\mathbb{P}^1$
  • Symbols
    • FindCharacter -- An option to compute the character instead of the dimension
    • GorensteinAlg -- An option to check the WLP for Gorenstein algebras
    • MonomialAlg -- An option to check the WLP for R/I with I Monomial ideal
    • Multidegree -- An option to compute the multidegrees arising from a torus action on $\mathbb{P}^1$
    • UseConjecture -- An option that, whenever the Han-Monsky is computed, allows the user to choose if employ Conjecture 4.1 [KMRR,25] or not

For the programmer

The object IncidenceCorrespondenceCohomology is a package, defined in IncidenceCorrespondenceCohomology.m2.

The source of this document is in IncidenceCorrespondenceCohomology.m2:1226:0.