pith. sign in

arxiv: 2605.30726 · v1 · pith:YZVKUGJBnew · submitted 2026-05-29 · 🧮 math.AG

The singular locus of a GL-variety

Christopher Heng Chiu , Alessandro Danelon , Andrew Snowden This is my paper

Pith reviewed 2026-06-28 20:52 UTC · model grok-4.3

classification 🧮 math.AG
keywords GL-varietysingular locusinfinite-dimensional varietygroup actionintrinsic characterizationalgebraic geometry
0
0 comments X

The pith

GL-varieties admit intrinsic characterizations of their singular locus that validate the prior candidate definition.

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

The paper supplies several descriptions of the singular locus on a GL-variety that refer only to the variety itself rather than to auxiliary finite-dimensional objects. These descriptions coincide with the candidate definition given in earlier work and thereby establish its correctness while making the geometric character of singular points more visible. A reader would care because GL-varieties arise naturally when studying infinite-dimensional representations and their invariants, and an intrinsic notion of singularity removes an extra layer of construction. If the characterizations hold, local questions about these varieties can be settled directly from the given data of X.

Core claim

A GL-variety X is typically an infinite-dimensional variety equipped with an action of the infinite general linear group. The paper gives a number of characterizations of the singular locus that are intrinsic to X. This work shows that the candidate definition is clearly correct, and helps clarify the geometric meaning of singular points.

What carries the argument

Intrinsic characterizations of the singular locus of a GL-variety X that match the auxiliary finite-dimensional construction.

If this is right

  • Singular points can be detected without first building the auxiliary finite-dimensional varieties.
  • The geometric meaning of a singular point becomes expressible in terms of the orbit or stabilizer data native to X.
  • The earlier candidate definition is confirmed as the right one for all GL-varieties.
  • Local geometric questions about GL-varieties can now be posed and answered directly from the intrinsic structure.

Where Pith is reading between the lines

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

  • The same intrinsic approach could be tested on varieties carrying actions of other infinite algebraic groups.
  • It may become possible to compute the singular locus for concrete families of GL-varieties by applying only the new tests.
  • Questions about resolution of singularities or deformation theory for these varieties could be rephrased intrinsically.

Load-bearing premise

The auxiliary finite-dimensional varieties used to define the candidate singular locus correctly capture the local geometry of the infinite-dimensional GL-variety at each point.

What would settle it

An explicit GL-variety together with a point where one of the new intrinsic tests declares the point singular while the auxiliary-variety test declares it nonsingular (or vice versa).

read the original abstract

A $\mathbf{GL}$-variety is a (typically) infinite dimensional variety $X$ equipped with an action of the infinite general linear group. In recent work of the first two authors with Draisma, a candidate definition for the singular locus of $X$ was put forth. The approach there made use of auxiliary finite dimensional varieties associated to $X$. In this paper, we give a number of characterizations of the singular locus that are intrinsic to $X$. This work shows that the candidate definition is clearly correct, and helps clarify the geometric meaning of singular points.

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

0 major / 2 minor

Summary. The paper introduces intrinsic characterizations of the singular locus of a GL-variety X (an infinite-dimensional variety with a GL-action) and proves their equivalence to the candidate definition from prior work that relied on auxiliary finite-dimensional varieties associated to X. The central claim is that these intrinsic descriptions confirm the correctness of the candidate definition and clarify the geometric meaning of singular points.

Significance. If the equivalences hold, the work provides a robust foundation for singularity theory on GL-varieties by eliminating dependence on auxiliary constructions, which strengthens the geometric interpretation and may facilitate further study of infinite-dimensional algebraic geometry. The multiple independent characterizations add value by allowing cross-verification.

minor comments (2)
  1. The abstract and introduction could more explicitly reference the specific prior work (with Draisma) by citation number or section to clarify the relationship between the candidate definition and the new intrinsic ones.
  2. Notation for the GL-action and the auxiliary varieties should be standardized across sections to avoid minor ambiguity in the equivalence statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The report accurately captures the contribution of the intrinsic characterizations of the singular locus of GL-varieties.

Circularity Check

0 steps flagged

No significant circularity; new intrinsic characterizations are independent of prior candidate definition

full rationale

The paper references prior overlapping-author work (Chiu-Danelon-Draisma) solely to introduce the candidate definition based on auxiliary finite-dimensional varieties. The core contribution consists of newly derived intrinsic characterizations of the singular locus for the GL-variety X, which are constructed directly from the geometry of X itself and then shown to coincide with the candidate. No equation, definition, or central claim in the provided abstract reduces the new characterizations to the prior definition by construction, nor does the argument rest exclusively on the self-citation. The derivation chain is therefore self-contained, with the intrinsic results supplying independent verification rather than a tautological restatement of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.1-grok · 5614 in / 980 out tokens · 21212 ms · 2026-06-28T20:52:13.420178+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 21 canonical work pages · 5 internal anchors

  1. [1]

    Eggermont, Andrew Snowden

    Arthur Bik, Jan Draisma, Rob H. Eggermont, Andrew Snowden. The geometry of polynomial representations. Int.\ Math.\ Res.\ Not.\ IMRN (2022). doi:10.1093/imrn/rnac220 arXiv:2105.12621

    work page doi:10.1093/imrn/rnac220 2022

  2. [2]

    Eggermont, Andrew Snowden

    Arthur Bik, Jan Draisma, Rob H. Eggermont, Andrew Snowden. Uniformity for limits of tensors. arXiv:2305.19866

    work page arXiv

  3. [3]

    Eggermont, Andrew Snowden

    Arthur Bik, Jan Draisma, Rob H. Eggermont, Andrew Snowden. Improved unirationalty for -varieties. In preparation

  4. [4]

    The geometry of polynomial representations in positive characteristic

    Arthur Bik, Jan Draisma, Andrew Snowden. The geometry of polynomial representations in positive characteristic. Math Z. 310 (2025). doi:10.1007/s00209-025-03720-y arXiv:2406.07415

    work page doi:10.1007/s00209-025-03720-y 2025

  5. [5]

    On the nilpotent functions at a non-degenerate arc

    David Bourqui, Mercedes Haiech. On the nilpotent functions at a non-degenerate arc. manuscripta math. 165 (2021). doi:10.1007/s00229-020-01209-y

    work page doi:10.1007/s00229-020-01209-y 2021

  6. [6]

    Smooth arcs on algebraic varieties

    David Bourqui, Julien Sebag. Smooth arcs on algebraic varieties. J.\ Sing. 16 (2017). doi:10.5427/jsing.2017.16f

    work page doi:10.5427/jsing.2017.16f 2017

  7. [7]

    The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space

    Christopher Chiu. The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space. arXiv:math/2504.05073

    work page internal anchor Pith review arXiv

  8. [8]

    Embedding codimension of the space of arcs

    Christopher Chiu, Tommaso de Fernex, Roi Docampo. Embedding codimension of the space of arcs. Forum Math.\ Pi 10 (2022). doi:10.1017/fmp.2021.19 arXiv:2001.08377

    work page doi:10.1017/fmp.2021.19 2022

  9. [9]

    On arc fibers of morphisms of schemes

    Christopher Chiu, Tommaso de Fernex, Roi Docampo. On arc fibers of morphisms of schemes. J. Eur.\ Math.\ Soc. (2024), published online first. doi:10.4171/JEMS/1500 arXiv:2206.08060

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4171/jems/1500 2024

  10. [10]

    Singular loci of varieties in polynomial functors

    Christopher Chiu, Alessandro Danelon, Jan Draisma. Singular loci of varieties in polynomial functors. arXiv:2501.07497

    work page arXiv

  11. [11]

    Sym-Noetherianity for powers of GL-varieties

    Christopher Chiu, Alessandro Danelon, Jan Draisma, Rob Eggermont, Azhar Farooq. Sym-Noetherianity for powers of GL-varieties. Algebra Number Theory 19 (2025), no. 11, pp. 2091--2118. doi:10.2140/ant.2025.19.2091 arXiv:2212.05790

    work page doi:10.2140/ant.2025.19.2091 2025

  12. [12]

    Stabilization for varieties in polynomial functors

    Alessandro Danelon. Stabilization for varieties in polynomial functors. Ph. D.\ thesis, Eindhoven University of Technology, 2023. \\ Available at: ://pure.tue.nl/ws/portalfiles/portal/268926250/20230217_Danelon_hf.pdf

    work page arXiv 2023

  13. [13]

    Topological Noetherianity of polynomial functors

    Jan Draisma. Topological Noetherianity of polynomial functors. J.\ Amer.\ Math.\ Soc. 32(3) (2019), pp.\ 691--707. doi:10.1090/jams/923 arXiv:1705.01419

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/jams/923 2019

  14. [14]

    Computation of point modules of finitely semi-graded rings.Comm

    Jan Draisma, Michal Laso\'n, Anton Leykin. Stillman’s conjecture via generic initial ideals. Comm.\ Algebra 47 (2019), pp. 2384--2395. doi:10.1080/00927872.2019.1574806 arXiv:1802.10139v2

    work page doi:10.1080/00927872.2019.1574806 2019

  15. [15]

    \'El\'ements de g\'eom\'etrie alg\'ebrique : IV

    Alexander Grothendieck. \'El\'ements de g\'eom\'etrie alg\'ebrique : IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas, Premi\'ere partie. Publ.\ Math.\ Inst.\ Hautes \'Etudes Sci. 20 (1964). \\ Available at: ://www.numdam.org/item/PMIHES_1964__20__5_0/

    1964

  16. [16]

    Jet schemes and singularities

    Lawrence Ein, Mircea Mustata. Jet schemes and singularities. arXiv:math/0612862

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Big polynomial rings and Stillman's conjecture

    Daniel Erman, Steven Sam, Andrew Snowden. Big polynomial rings and Stillman's conjecture. Invent.\ Math. 218 (2019), no. 2, pp. 413--439. doi:10.1007/s00222-019-00889-y arXiv:1801.09852

    work page doi:10.1007/s00222-019-00889-y 2019

  18. [18]

    Ultrahomogeneous tensor spaces

    Nate Harman, Andrew Snowden. Ultrahomogeneous tensor spaces. Adv.\ Math. 443 (2024). \\ doi:10.1016/j.aim.2024.109599 arXiv:2207.09626

    work page doi:10.1016/j.aim.2024.109599 2024

  19. [19]

    Tensor spaces and the geometry of polynomial representations

    Nate Harman, Andrew Snowden. Tensor spaces and the geometry of polynomial representations. arXiv:2407.19132

    work page arXiv

  20. [20]

    Flatness of tangent cones of a family of hypersurfaces

    Gary Kennedy. Flatness of tangent cones of a family of hypersurfaces. doi:10.1016/0021-8693(90)90051-O

    work page doi:10.1016/0021-8693(90)90051-o

  21. [21]

    Stable representation theory: beyond the classical groups

    Andrew Snowden. Stable representation theory: beyond the classical groups. arXiv:2109.11702

    work page arXiv

  22. [22]

    ://stacks.math.columbia.edu (accessed 2026)

    Stacks Project. ://stacks.math.columbia.edu (accessed 2026)

    2026

  23. [23]

    Tanimoto

    H. Tanimoto. Some characterizations of smoothness. J.\ Math.\ Kyoto Univ. 23 (1983), no. 4, 695--706. doi:10.1215/kjm/1250521430

    work page doi:10.1215/kjm/1250521430 1983

  24. [24]

    Jets via Hasse-Schmidt Derivations

    Paul Vojta. Jets via Hasse--Schmidt derivations. arXiv:math/0407113

    work page internal anchor Pith review Pith/arXiv arXiv