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Mathematics > Algebraic Geometry

arXiv:2208.06299 (math)
[Submitted on 12 Aug 2022 (v1), last revised 10 Jan 2023 (this version, v3)]

Title:Hessenberg varieties of codimension one in the flag variety

Authors:Laura Escobar, Martha Precup, John Shareshian
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Abstract:We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincaré polynomial, (2) characterize when these varieties are irreducible, and (3) show that all are reduced schemes. We prove that the singular locus of any nilpotent codimension one Hessenberg variety is also a Hessenberg variety. A key tool in our analysis is a new result applying to all (type A) Hessenberg varieties without any restriction on codimension, which states that their Poincaré polynomials can be computed by counting the points in the corresponding variety defined over a finite field. The results below were originally motivated by work of the authors in [arXiv:2107.07929] studying the precise relationship between Hessenberg and Schubert varieties, and we obtain a corollary extending the results from that paper to all codimension one (type A) Schubert varieties.
Comments: Version 2 includes updates to the statement of Theorem 5, its proof, and the example appearing at the end of Section 5
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14M17, 14M15
Cite as: arXiv:2208.06299 [math.AG]
  (or arXiv:2208.06299v3 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2208.06299

Submission history

From: Martha Precup [view email]
[v1] Fri, 12 Aug 2022 14:34:23 UTC (25 KB)
[v2] Tue, 16 Aug 2022 21:49:15 UTC (25 KB)
[v3] Tue, 10 Jan 2023 15:25:23 UTC (25 KB)
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