pith. sign in

arxiv: 2606.03683 · v1 · pith:NJ574M7Rnew · submitted 2026-06-02 · 🧮 math.AG

Improved unirationality for GL-varieties

Arthur Bik , Jan Draisma , Rob Eggermont , Andrew Snowden This is my paper

Pith reviewed 2026-06-28 08:23 UTC · model grok-4.3

classification 🧮 math.AG
keywords GL-varietyunirationalitysurjective morphismsecant varietytensor varietyinfinite-dimensional varietygeneral linear group
0
0 comments X

The pith

Irreducible GL-varieties admit surjective GL-equivariant maps from a simple product model rather than only dominant maps.

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

The paper strengthens the existing unirationality theorem by proving that the dominant map from a simple GL-variety onto an irreducible GL-variety can always be upgraded to a surjective map. A GL-variety carries an action of the infinite general linear group and is typically infinite-dimensional. The simple model is the product of an irreducible finite-dimensional variety with trivial action and an infinite-dimensional affine space with linear action. This upgrade applies directly to secant varieties of tensor varieties, which are therefore images of GL-equivariant maps rather than closures of such images. The paper derives several further consequences from the improved statement.

Core claim

If an irreducible GL-variety satisfies the hypotheses of the unirationality theorem, then there exists a surjective GL-equivariant morphism from the product of an irreducible finite-dimensional variety with trivial GL-action and an infinite-dimensional affine space with linear GL-action onto the given GL-variety. This replaces the merely dominant map of the earlier theorem. As an immediate consequence, secant varieties to varieties of tensors are images of GL-equivariant maps.

What carries the argument

The surjective GL-equivariant morphism from the product of an irreducible finite-dimensional variety with trivial action and an infinite-dimensional affine space with linear action.

Load-bearing premise

The GL-variety must be irreducible and satisfy the hypotheses under which the earlier unirationality theorem supplies a dominant map.

What would settle it

An explicit irreducible GL-variety equipped with a point that lies outside the image of every GL-equivariant map from the product of a finite-dimensional irreducible variety with trivial action and an infinite-dimensional affine space with linear action.

read the original abstract

A $\mathbf{GL}$-variety is a typically infinite dimensional variety equipped with a suitable action of the infinite general linear group $\mathbf{GL}$. In earlier work, we established the unirationality theorem: an irreducible $\mathbf{GL}$-variety admits a dominant map from a particularly simple $\mathbf{GL}$-variety, namely, the product of an irreducible finite-dimensional variety with trivial $\mathbf{GL}$-action and an infinite-dimensional affine space on which $\mathbf{GL}$ acts linearly. The main result of this paper states that this map can in fact be constructed to be surjective rather than merely dominant. An immediate application is that secant varieties to varieties of tensors, which are typically constructed as image closures of certain $\mathbf{GL}$-equivariant maps, are in fact also images of (more complicated) $\mathbf{GL}$-equivariant maps. We derive several consequences of this improved unirationality theorem.

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 / 1 minor

Summary. The manuscript improves the unirationality theorem for irreducible GL-varieties: under the same hypotheses as the prior result (which produced a dominant GL-equivariant map), there exists a surjective GL-equivariant map from the product of an irreducible finite-dimensional variety with trivial GL-action and an infinite-dimensional affine space with linear GL-action. The paper supplies an explicit modification of the earlier construction to achieve surjectivity while preserving equivariance, applies the result to show that secant varieties of tensor varieties are images (not merely closures of images) of such maps, and derives several consequences.

Significance. If the central construction holds, the result strengthens the geometric understanding of GL-varieties by replacing dominance with surjectivity, which is a meaningful improvement for applications such as the geometry of secant varieties. The explicit modification and direct consequences for tensor secants constitute a clear technical advance within the existing framework.

minor comments (1)
  1. The abstract and introduction would benefit from a brief pointer to the precise section containing the modified construction that upgrades dominance to surjectivity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report accurately captures the main result and its applications.

Circularity Check

0 steps flagged

Minor self-citation to prior dominant-map theorem; new surjectivity construction is independent

full rationale

The paper cites the authors' earlier unirationality theorem (providing a dominant GL-equivariant map) and supplies an explicit modification of that construction to achieve surjectivity while preserving equivariance and the same hypotheses on the target GL-variety. No step reduces the new claim to the prior result by definition, renaming, or fitted input; the argument is a direct mathematical improvement. This matches the expected pattern of a self-contained derivation with non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the prior unirationality theorem and standard definitions of GL-varieties and irreducibility; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Standard axioms of algebraic geometry over an algebraically closed field, including properties of morphisms and group actions.
    Invoked implicitly in the definitions of GL-varieties and dominant/surjective maps.
  • domain assumption The earlier unirationality theorem providing a dominant map from the simple GL-variety.
    The new result builds directly on this prior theorem as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5687 in / 1175 out tokens · 20736 ms · 2026-06-28T08:23:39.924644+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

14 extracted references · 12 canonical work pages · 4 internal anchors

  1. [1]

    Small Subalgebras of Polynomial Rings and Stillman's Conjecture

    Tigran Ananyan, Melvin Hochster. Small subalgebras of polynomial rings and Stillman's conjecture. J. Amer. Math. Soc. 33 (2020), no. 1, pp. 2910--309. doi:10.1090/jams/932 arXiv:1610.09268v1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/jams/932 2020

  2. [2]

    Strength and slice rank of forms are generically equal

    Edoardo Ballico, Arthur Bik, Alessandro Oneto, Emanuele Ventura. Strength and slice rank of forms are generically equal. Israel J. Math. (2022). doi:10.1007/s11856-022-2397-0 arXiv:2102.11549

    work page doi:10.1007/s11856-022-2397-0 2022

  3. [3]

    The set of forms with bounded strength is not closed

    Edoardo Ballico, Arthur Bik, Alessandro Oneto, Emanuele Ventura. The set of forms with bounded strength is not closed. Compt. Rend. Math. 360 (2022), pp. 371--380. doi:10.5802/crmath.302 arXiv:2012.01237

    work page doi:10.5802/crmath.302 2022

  4. [4]

    Eggermont, Andrew Snowden

    Arthur Bik, Jan Draisma, Rob H. Eggermont, Andrew Snowden. The geometry of polynomial representations. Int.\ Math.\ Res.\ Not.\ IMRN 16 (2023), pp. 14131--14195. doi:10.1093/imrn/rnac220 arXiv:2105.12621

    work page doi:10.1093/imrn/rnac220 2023

  5. [5]

    Eggermont, Andrew Snowden

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

    work page arXiv

  6. [6]

    The geometry of polynomial representations in positive characteristic

    Arthir Bik, Jan Draisma, Andrew Snowden. The geometry of polynomial representations in positive characteristic. Math. Z. 310(1) (2025), paper no. 13, 39 pp. doi:10.1007/s00209-025-03720-y arXiv:2406.07415

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

  7. [7]

    Topological noetherianity for cubic polynomials

    Harm Derksen, Rob H. Eggermont, Andrew Snowden. Topological noetherianity for cubic polynomials. Algebra Number Theory 11 (2017), no. 9, pp. 2197--2212. doi:10.2140/ant.2017.11.2197 arXiv:1701.01849

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/ant.2017.11.2197 2017

  8. [8]

    Topological Noetherianity of polynomial functors

    Jan Draisma. Topological Noetherianity of polynomial functors. J.\ Amer.\ Math.\ Soc. 32 (2019), no. 3, 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

  9. [9]

    Applications of algebraic combinatorics to algebraic geometry

    David Kazhdan, Tamar Ziegler. Applications of algebraic combinatorics to algebraic geometry. Indag.\ Math., New Ser. 32 (2021), no. 6, pp. 1412--1428. doi:10.1016/j.indag.2021.09.002 \ :2005.12542

    work page doi:10.1016/j.indag.2021.09.002 2021

  10. [10]

    Landsberg

    Joseph M. Landsberg. Tensors: geometry and applications. Graduate Studies in Mathematics 128, American Mathematical Society, Providence, RI, 2012

    2012

  11. [11]

    Noetherianity of some degree two twisted commutative algebras

    Rohit Nagpal, Steven V Sam, Andrew Snowden. Noetherianity of some degree two twisted commutative algebras. Selecta Math.\ (N.S.) 22 (2016), no. 2, pp. 913--937. doi:10.1007/s00029-015-0205-y arXiv:1501.06925

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00029-015-0205-y 2016

  12. [12]

    W. M. Schmidt. The density of integer points on homogeneous varieties. Acta Math. 154 (1985), no. 3--4, pp. 243--296. doi:10.1007/BF02392473

    work page doi:10.1007/bf02392473 1985

  13. [13]

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

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

    2026

  14. [14]

    Wadsworth

    Adrian R. Wadsworth. Hilbert subalgebras of finitely generated algebras. J.\ Algebra 43 (1976), pp. 298--304. doi:10.1016/0021-8693(76)90161-7

    work page doi:10.1016/0021-8693(76)90161-7 1976