Papers
Computing the cohomology of line bundles on the incidence correspondence and related invariants - Preprint 2025
with Annet Kyomuhangi, Claudiu Raicu, and Ethan Reed
Cohomology on the incidence correspondence and related questions - Preprint 2024
with Annet Kyomuhangi, Claudiu Raicu, and Ethan Reed
The cohomology of line bundles on flag varieties is well understood over fields of characteristic zero, thanks to the Borel-Weil-Bott theorem, but remains largely open in positive characteristic. In this paper, we study the cohomology of line bundles on the incidence correspondence, the partial flag variety parameterizing pairs consisting of a point in projective space and a hyperplane containing it. We establish a recursive formula for the characters of these cohomology groups in positive characteristic and, in characteristic 2, a non-recursive description in terms of truncated Schur and Nim-symmetric polynomials.
We also highlight surprising connections to other questions of interest in commutative algebra, such as the graded Han-Monsky representation ring and the Weak Lefschetz Property for Artinian monomial complete intersections.
The non-Lefschetz locus for conics - Preprint 2024
In this paper, we prove that any complete intersection A=k[x1,x2,x3]/(f1,f2,f3), in characteristic zero, has the Strong Lefschetz Property at range 2, i.e. there exists a linear form ℓ∈[R]1, such that the multiplication map ×ℓ²:[A]i→[A]i+2 has maximum rank in each degree.
We then study the forms of degree 2 for which the map ×C:[A]i→[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[x1,x2,x3]/(f1,f2,f3) has the expected codimension as a subscheme of ℙ⁵.
Finally, to extend a similar result to the first cohomology modules of rank 2 vector bundles over ℙ², we explore the connection between non-Lefschetz conics and jumping conics. The non-Lefschetz locus of conics is a subset of the jumping conics, which, unlike the case of lines, can be a proper subset when E is semistable with even first Chern class.
The non-Lefschetz locus for vector bundle of rank 2 on ℙ² - Journal of Algebra 2023
A finite length graded R-module M has the Weak Lefschetz Property if there is a linear form l in R such that the multiplication map xl from Mi to Mi+1 has maximal rank. The set of linear forms with this property forms 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 of a rank 2 vector bundle E over ℙ². The main result is that this non-Lefschetz locus has the expected codimension under the assumption that E is general.
Some notes and corrections of the paper “The Non-Lefschetz locus” - Journal of Algebra 2023
Selected Talks
04/15/2025 - KTH, Sweden
Problem-solving seminar in commutative algebra
Lefschetz Properties for Artinian Complete Intersections
03/17/2025 - University of Notre Dame
Algebraic Geometry and Commutative Algebra Seminar
Cohomology of line bundles on the incidence correspondence
03/17/2025 - Field Institute, Toronto
Commutative Algebra and Applications Seminar
Cohomology of line bundles on the incidence correspondence
04/21/2024 - AMS meeting at University of Texas, San Antonio
Special Session on "Commutative Algebra and connections to combinatorics”
Lefschetz Properties and non-Lefschetz locus
04/21/2024 - AMS meeting, University of Wisconsin-Milwaukee
Special Session on "Ramification in Algebraic and Arithmetic Geometry"
Lefschetz Properties and non-Lefschetz locus
11/05/2023 - Arizona State University
Conference Algebra Days at Arizona State
The non-Lefschetz locus, jumping lines and conics
10/07/2023 - AMS meeting in Omaha, Nebraska
Special Session on "Varieties with Unexpected Hypersurfaces, Geproci Sets and their Interactions"
The non-Lefschetz locus for lines and for conics
09/18/2023 - University of Illinois Chicago
Algebraic Geometry Seminar
The non-Lefschetz locus, jumping lines and conics
05/16/2023 - Fields Institute, Toronto
Workshop on Lefschetz Properties in Algebra, Geometry, Topology and Combinatorics
The non-Lefschetz locus for lines and for conics
05/07/2023 - University of Nebraska-Lincoln
URiCA, Upcoming Researchers in Commutative Algebra Conference
The non-Lefschetz locus of the first cohomology module of a locally free sheaf E of rank 2 over ℙ²
01/06/2023 - JMM 2023, Boston
Special session "The combinatorics and geometry of Jordan type and Lefschetz properties"