Schubert varieties for the super affine Grassmannian of $GL_{n|m}$

Sofia Gureva; Vladimir Stukopin

arxiv: 2606.17318 · v1 · pith:TUOODALMnew · submitted 2026-06-15 · 🧮 math.AG

Schubert varieties for the super affine Grassmannian of GL_(n|m)

Vladimir Stukopin , Sofia Gureva This is my paper

Pith reviewed 2026-06-27 02:09 UTC · model grok-4.3

classification 🧮 math.AG

keywords super affine GrassmannianSchubert varietiesGL_{n|m}supervarietiesaffine Grassmanniansupergroupsalgebraic geometry

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The pith

The Schubert varieties of the super affine Grassmannian for GL_{n|m} admit explicit coordinate descriptions and polynomial equations in the cases n|m = 1|1 and 2|1.

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

This paper conducts an explicit computational study of Schubert varieties on the affine Grassmannian associated to the general linear supergroup GL_{n|m}. It focuses on the low-dimensional cases of 1|1 and 2|1, where it provides concrete coordinate charts, defining equations, and morphisms that capture the supervariety structures. A sympathetic reader would care because these descriptions offer the first concrete examples of how supergeometry manifests in affine Grassmannians and their Schubert cells. Understanding these small cases could illuminate the behavior in higher dimensions or in related supergeometric constructions. The work demonstrates that direct computation is feasible when the dimensions are small enough for explicit affine charts.

Core claim

In the dimensions n|m = 1|1 and 2|1, the Schubert varieties in the super affine Grassmannian of GL_{n|m} are cut out by specific polynomial equations in explicit affine coordinate charts, and the morphisms between them can be written down explicitly, revealing the underlying supervariety structure.

What carries the argument

The explicit affine coordinate chart on the super affine Grassmannian, in which the Schubert varieties are defined by polynomial equations respecting the superalgebra structure.

If this is right

The supervariety structures of these Schubert varieties can be written down with both even and odd coordinates.
Morphisms between the varieties are given explicitly in the chosen coordinates.
Direct computation of the equations is possible in these low dimensions due to the existence of suitable affine charts.
The descriptions provide concrete models for the intersection theory or cohomology in the super setting for these cases.

Where Pith is reading between the lines

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

The pattern of equations found here may suggest a recursive or combinatorial description that extends to higher n|m if similar charts can be constructed.
These explicit examples could be used to verify or refute general conjectures about the dimension or smoothness of Schubert varieties in super affine Grassmannians.
The computational approach might connect to the study of super flag varieties or representations of supergroups where similar coordinate methods apply.

Load-bearing premise

The super affine Grassmannian admits an explicit affine coordinate chart in which Schubert varieties are cut out by polynomial equations that can be computed directly for these small dimensions.

What would settle it

Attempting to derive the defining equations in an affine chart for the case n|m = 3|1 and finding that no consistent set of super-polynomial equations defines a variety with the expected dimension or super-structure.

read the original abstract

We study Schubert varieties of the affine Grassmannian for the general linear supergroup $GL_{n|m}$. An explicit computational study is conducted in low-dimensional cases, namely for dimensions $n|m = 1|1$ and $2|1$. We describe the supervariety structures that arise in these settings, providing coordinate descriptions, equations, and morphisms.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit coordinate descriptions, equations, and morphisms for Schubert varieties on the super affine Grassmannian in the two smallest cases, 1|1 and 2|1.

read the letter

The paper carries out direct, low-dimensional computations of Schubert varieties inside the super affine Grassmannian for GL_{n|m}. It supplies coordinate charts, the actual equations that cut out the varieties, and some morphisms for the cases n|m = 1|1 and 2|1.

That explicitness is the main positive. In supergeometry the odd directions often complicate the picture, so having the supervariety structures written down for these tiny dimensions gives people something concrete to check or build on. The approach uses standard affine charts and translates Schubert conditions into polynomial (or super-polynomial) equations, which the stress-test note says is reproducible by hand or software in these sizes. No general theorem or limit argument is being invoked, so the claim stays within what direct computation can support.

The obvious limitation is the narrow scope. These are the smallest possible nontrivial dimensions, so the work stays case-by-case and does not produce a general description or new structural result. If the goal is to understand how super Schubert varieties behave in general, this paper only supplies two data points. The abstract does not indicate any attempt to push the method further or compare with classical results in a systematic way.

Specialists already working on super Grassmannians or super Schubert calculus will find the coordinate data useful as examples. A broader audience will not. The computations look like honest, verifiable work rather than circular or invented constructions. I would send the paper to peer review so that the explicit equations can be checked by people who know the super conventions; the bar for this kind of concrete case study is not high, and the paper meets it.

Referee Report

0 major / 2 minor

Summary. The manuscript conducts an explicit computational study of Schubert varieties in the super affine Grassmannian of GL_{n|m}, focusing on the low-dimensional cases n|m = 1|1 and 2|1. It supplies coordinate descriptions, defining equations, and morphisms that realize the supervariety structures in these settings.

Significance. If the explicit descriptions are accurate, the work supplies concrete, reproducible examples in super algebraic geometry that can serve as test cases for broader conjectures on Schubert varieties in the super setting. The direct-computation approach in finite, small dimensions is methodologically appropriate and avoids reliance on unverified general identities.

minor comments (2)

The abstract and introduction should include a brief recall of the standard definition of the (super) affine Grassmannian and the Schubert conditions used, to make the coordinate computations self-contained for readers outside the immediate subfield.
Notation for the super coordinates (even and odd variables) and the precise super-ring in which the defining equations live should be stated uniformly at the beginning of each case study.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, assessment of significance, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit low-dimensional computations are self-contained

full rationale

The paper performs direct, explicit computations of coordinate charts, equations, and morphisms for the small cases n|m=1|1 and 2|1 using standard definitions of GL_{n|m} and its affine Grassmannian. No derivation step reduces by construction to a fitted input, self-definition, or self-citation chain; the results are presented as reproducible hand or symbolic computations without invoking general theorems whose validity depends on the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5577 in / 935 out tokens · 28497 ms · 2026-06-27T02:09:13.843283+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 canonical work pages

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Pith/arXiv arXiv

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2013 , publisher=

Gauge Field Theory and Complex Geometry , author=. 2013 , publisher=

2013

[2] [2]

2011 , publisher=

Mathematical foundations of supersymmetry , author=. 2011 , publisher=

2011

[3] [3]

On super Plücker embedding and cluster algebras , volume=

Shemyakova, Ekaterina and Voronov, Theodore , year=. On super Plücker embedding and cluster algebras , volume=. Selecta Mathematica , publisher=. doi:10.1007/s00029-021-00756-w , number=

work page doi:10.1007/s00029-021-00756-w

[4] [4]

On affine Grassmannians, shifted Yangians, and W-algebras in the super-case (Master thesis) , author=

[5] [5]

Contemporary Mathematics , volume=

Quotients in supergeometry , author=. Contemporary Mathematics , volume=

[6] [6]

Journal of mathematical physics , volume=

The Minkowski and conformal superspaces , author=. Journal of mathematical physics , volume=. 2007 , publisher=

2007

[7] [7]

2004 , publisher=

Supersymmetry for mathematicians: an introduction , author=. 2004 , publisher=

2004

[8] [8]

The Affine Grassmannian with a View Towards Geometric Satake , author=

[9] [9]

Jet Schemes and Their Applications in Singularities, Toric Resolutions and Integer Partitions

Mourtada, Hussein. Jet Schemes and Their Applications in Singularities, Toric Resolutions and Integer Partitions. Handbook of Geometry and Topology of Singularities IV. 2023

2023

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Geometric Langlands duality and representations of algebraic groups over commutative rings

Mirkovi \'c , Ivan and Vilonen, Kari. Geometric Langlands duality and representations of algebraic groups over commutative rings. Annals of mathematics. 2007

2007

[11] [11]

Mirabolic Satake equivalence and supergroups , volume=

Braverman, Alexander and Finkelberg, Michael and Ginzburg, Victor and Travkin, Roman , year=. Mirabolic Satake equivalence and supergroups , volume=. Compositio Mathematica , publisher=. doi:10.1112/s0010437x21007387 , number=

work page doi:10.1112/s0010437x21007387

[12] [12]

arXiv preprint arXiv:1603.05593 , year=

An introduction to affine Grassmannians and the geometric Satake equivalence , author=. arXiv preprint arXiv:1603.05593 , year=

Pith/arXiv arXiv