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arxiv: 2208.06299 · v4 · submitted 2022-08-12 · 🧮 math.AG

Hessenberg varieties of codimension one in the flag variety

Laura Escobar , Martha Precup , John Shareshian This is my paper

Pith reviewed 2026-05-24 12:00 UTC · model grok-4.3

classification 🧮 math.AG
keywords Hessenberg varietiesflag varietyPoincaré polynomialreduced schemestype ASchubert varietiesnilpotent elements
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The pith

Hessenberg varieties of codimension one are reduced schemes whose Poincaré polynomials equal their finite-field point counts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines Hessenberg varieties of codimension one inside the type A flag variety. It supplies an explicit formula for their Poincaré polynomials, determines when they are irreducible, and establishes that they are always reduced schemes. A general tool shows that the Poincaré polynomial of any type A Hessenberg variety equals the number of points of the corresponding variety over a finite field, and this identification is applied without extra geometric restrictions. The singular locus of any nilpotent codimension one example is itself a Hessenberg variety. The work extends earlier comparisons of Hessenberg varieties with Schubert varieties to the full codimension one case.

Core claim

The authors prove that the Poincaré polynomial of every type A Hessenberg variety equals the number of points over a finite field. For the codimension one case this yields an explicit formula, proves the varieties are reduced, gives a characterization of irreducibility, and shows that the singular locus of a nilpotent codimension one Hessenberg variety is again a Hessenberg variety. The point-counting identification requires no smoothness or special nilpotency hypotheses.

What carries the argument

The result that the Poincaré polynomial of any type A Hessenberg variety equals the number of its points over a finite field.

If this is right

  • An explicit formula for the Poincaré polynomial is obtained for the codimension one case.
  • Irreducibility of these varieties can be characterized.
  • All codimension one Hessenberg varieties are reduced schemes.
  • The singular locus of any nilpotent codimension one example is a Hessenberg variety.
  • Earlier results relating Hessenberg and Schubert varieties extend to all codimension one Schubert varieties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The point-counting method supplies a uniform computational route to Betti numbers for every type A Hessenberg variety.
  • Reducedness in codimension one may guide the study of scheme-theoretic properties for Hessenberg varieties in other codimensions.

Load-bearing premise

The point-counting result for Poincaré polynomials applies to arbitrary type A Hessenberg varieties without smoothness or other extra conditions.

What would settle it

A direct computation of the Poincaré polynomial for one Hessenberg variety that fails to match its point count over a finite field.

read the original 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 Poincar\'e 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 Poincar\'e 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript studies Hessenberg varieties of codimension one in the type A flag variety. It claims three main results: a formula for the Poincaré polynomial obtained by counting points over finite fields, a characterization of irreducibility, and a proof that all such varieties are reduced schemes. It further shows that the singular locus of any nilpotent codimension-one Hessenberg variety is itself a Hessenberg variety. The central tool is a new theorem, stated to apply to all type A Hessenberg varieties without codimension restriction, equating their Poincaré polynomials to the number of F_q-points; this is used to extend results from the authors' prior work on the relationship between Hessenberg and Schubert varieties.

Significance. If the general point-counting theorem holds without hidden smoothness or reducedness hypotheses, it supplies a practical method for computing Poincaré polynomials of arbitrary type A Hessenberg varieties. The codimension-one results on reducedness, irreducibility, and singular loci would then give concrete geometric information and a clean extension of the earlier Schubert-variety comparison.

major comments (1)
  1. [Abstract and general theorem] Abstract (and the statement of the general point-counting theorem): the claim that the Poincaré polynomial equals the F_q-point count for arbitrary type A Hessenberg varieties, with no codimension restriction and without smoothness or nilpotency hypotheses, is load-bearing for all three main results. The equality typically requires an affine cell decomposition or pure-weight étale cohomology with Frobenius eigenvalues q^{i/2}; the manuscript must explicitly verify that these conditions hold in the general case or identify the precise hypotheses under which the theorem applies.
minor comments (1)
  1. [Introduction] The abstract states that the results extend those of arXiv:2107.07929 to all codimension-one Schubert varieties; the introduction should include a precise statement of the corollary and a short comparison with the earlier paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a key point regarding the hypotheses of the general point-counting theorem. We address the major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

  1. Referee: [Abstract and general theorem] Abstract (and the statement of the general point-counting theorem): the claim that the Poincaré polynomial equals the F_q-point count for arbitrary type A Hessenberg varieties, with no codimension restriction and without smoothness or nilpotency hypotheses, is load-bearing for all three main results. The equality typically requires an affine cell decomposition or pure-weight étale cohomology with Frobenius eigenvalues q^{i/2}; the manuscript must explicitly verify that these conditions hold in the general case or identify the precise hypotheses under which the theorem applies.

    Authors: We agree that the general point-counting theorem is central and that its applicability requires explicit justification of the underlying conditions. The manuscript's proof of this theorem proceeds by establishing an affine paving for all type A Hessenberg varieties (via a generalization of the standard Schubert cell decomposition adapted to the Hessenberg condition), which directly implies that the étale cohomology is pure of weight i in degree 2i and that the Poincaré polynomial equals the F_q-point count. However, the current write-up does not spell out this paving argument in sufficient detail for the unrestricted case. In the revision we will add a dedicated subsection (or appendix) that verifies the affine cell decomposition for arbitrary type A Hessenberg varieties, thereby making the hypotheses of the theorem fully explicit and addressing the referee's concern. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for motivation; new point-counting theorem presented as independent

full rationale

The paper introduces a new result equating Poincaré polynomials of arbitrary type A Hessenberg varieties to their F_q point counts, explicitly without codimension restriction or additional hypotheses like smoothness. This is described as a key tool in the analysis rather than a consequence of the cited prior work arXiv:2107.07929, which is referenced only for motivation and a corollary extending Schubert variety results. No equations, definitions, or self-citations are shown to reduce the central claims to inputs by construction, and the codimension-one results on reducedness and irreducibility build on the new theorem. This qualifies as at most a minor non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the point-counting claim implicitly relies on unstated background results from algebraic geometry and finite fields.

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Reference graph

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