Papers
Cohomology on the incidence correspondence and related questions - Preprint
with Annet Kyomuhangi, Claudiu Raicu, and Ethan Reed
We study the cohomology of line bundles on the incidence correspondence. 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.
The non-Lefschetz locus for conics - Preprint
In this paper, we prove that any complete intersection A=k[x1,x2,x3]/(f1,f2,f3), with char k=0, 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.
Then we focus on the forms of degree 2 such that ×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. 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 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. Unlike the case of the lines, this can be proper when E is semistable with first Chern class even.
The non-Lefschetz locus for vector bundle of rank 2 on ℙ² - Journal of Algebra
A finite length graded R-module M has the Weak Lefschetz Property if there is a linear element 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, I focus on the study of the non-Lefschetz locus for the first cohomology module of a locally free sheaf E of rank 2 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
Talks
09/14/2024
AMS meeting at University of Texas, San Antonio, Special Session on "Commutative Algebra and connections to combinatorics”
Lefschetz Properties and non-Lefschetz locus
05/04/2024
CAAGTUS 2024, University of Arizona
Lefschetz Properties and non-Lefschetz locus
04/21/2024
AMS meeting at University of Wisconsin-Milwaukee, Special Session on "Ramification in Algebraic and Arithmetic Geometry"
Lefschetz Properties and non-Lefschetz locus
11/05/2023
Algebra Days at Arizona State, Arizona State University
The non-Lefschetz locus, jumping lines and conics
10/07/2023
AMS meeting in Omaha, 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
Workshop on Lefschetz Properties in Algebra, Geometry, Topology and Combinatorics at Fields Institute, Toronto
The non-Lefschetz locus for lines and for conics
05/07/2023
URiCA, Upcoming Researchers in Commutative Algebra at the University of Nebraska-Lincoln
The non-Lefschetz locus of the first cohomology module of a locally free sheaf E of rank 2 over ℙ²
01/06/2023
Special session "The combinatorics and geometry of Jordan type and Lefschetz properties"
Joint Mathematics Meetings 2023, Boston