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The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity

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Pith's one-line read This paper proves that the worst destabilizing one-parameter subgroup for a toric rational curve with one unibranch singularity becomes, for all sufficiently large embedding dimensions, a fixed head followed by an explicit linear tail.

desk verdict A real persistence theorem for worst destabilizing 1-PS for toric rational curves, with a load-bearing but likely repairable gap in the torus-reduction step. read the letter →

arxiv 2502.00458 v1 pith:YDGXXLVQ submitted 2025-02-01 math.AG

Joshua Jackson , David Swinarski This is my paper · ORCID
classification math.AG MSC 14L2414H1052B20
keywords geometricinvarianttheoryworstdestabilizing1-parametersubgroupHilbert-MumfordfunctionChowpolytopetoricrationalcurveunibranchsingularitynumericalsemigroupconvexoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks, for a GIT-unstable Chow point of a toric rational curve with one unibranch singularity, which one-parameter subgroup is the worst destabilizer in Kempf's sense. It translates the Hilbert-Mumford ratio maximization into an explicit convex optimization problem: find the closest point on a polyhedral cone $W$ to a vector $a$ built from the singularity's numerical semigroup. The main theorem states that once the embedding dimension $N$ is large enough, the worst 1-PS stabilizes: its first $\ell$ weights are independent of $N$, and the remaining weights lie on an explicit line with slope $m=-6/((N-\ell+1)(N-\ell+2))$ and intercept $b=2+2(N+2\ell-1)/((N-\ell+1)(N-\ell+2))$. If correct, this gives a concrete asymptotic description of the instability of these singular curves, a step toward understanding moduli spaces of unstable objects. The proof works by solving a Karush-Kuhn-Tucker system on each face of the cone and showing the winning face is the same for all large $N$.

What carries the argument

The load-bearing object is the polyhedral cone $W\subset \mathbb{R}^{N+1}$ of convex weight vectors, whose facets force the slopes of the piecewise-linear weight graph to increase; a worst 1-PS must lie on $W$ because any non-convex or negative weight vector can be improved. The vector $a$, with coordinates $a_1=\gamma_1$, $a_i=\gamma_i-\gamma_{i-2}$ for $2\le i\le N$, and $a_{N+1}=1$ (or 2 in the Simplified Problem), packages the semigroup data so that the Hilbert-Mumford value equals $a\cdot w/\|w\|$, making the worst 1-PS the nearest point (proximum) from $a$ to $W$. The proof analyzes this proximum face by face through the KKT matrix equation $Ax=2a$, whose solutions are rational functions of $N$; Cramer-rule formulas for the determinants show the last coordinates satisfy polynomial recurrences, and the sign of the polynomials $\chi$ and $\psi$ decides which face carries the global optimum for large $N$.

What would settle it

Take a numerical semigroup not in the paper's tables, such as $\langle 3,4\rangle$, compute the proximum of $a$ to $W$ for a sequence of $N$ far beyond the predicted $N_0$, and check whether the tail equals $mi+b$; or directly test the $\mathbb{G}_m$ version of the Kempf-Morrison Lemma by searching for a non-torus 1-PS with larger $\mu/\|\lambda\|$ in a small example. A single counterexample to either check would show the paper's theorem, or its reduction step, is false.

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Extended reading notes

Core claim

The central claim is that the worst destabilizing 1-PS for the Chow point of $C_\Gamma$ persists in a precise sense: there exist integers $N_0$ and $\ell$, depending only on the semigroup $\Gamma$, such that for all $N \ge N_0$ the optimal weight vector $w^*$ has its first $\ell$ coordinates constant and its remaining coordinates equal to $w^*_i = m i + b$ with $m$ and $b$ as above. The proof first establishes the analogous persistence for a Simplified Problem, where the last entry of the target vector is 2 instead of 1, then constructs the unsimplified solution by adding a corner and using a least-squares regression line through the tail points $(\gamma_\ell, 2),\dots,(\gamma_{N-1},2),(\gamma_N,1)$. The authors verify the result by explicit KKT solutions for the faces of $W$, and give example tables for semigroups $\langle 2,3\rangle$, $\langle 2,5\rangle$, $\langle 4,9\rangle$, $\langle 5,7\rangle$, and $\langle 8,13\rangle$, plus a closed-form description for cusps $y^2=x^{2r+1}$.

Load-bearing premise

The reduction from all one-parameter subgroups to those in the maximal torus relies on an unproved extension of the Kempf-Morrison Lemma from finite automorphism groups to the infinite group $\mathbb{G}_m$ acting with distinct weights; if that extension fails, the convex problem solves only the torus-optimal 1-PS.

Editorial extensions

If this is right

  • For every numerical semigroup $\Gamma$, the worst 1-PS for all sufficiently large embedding dimensions is completely determined by a finite set of corner indices $I$ and the explicit linear tail; no further computation is needed once $N_0$ and $\ell$ are known.
  • The same KKT machinery gives a proof of persistence for the Simplified Problem, which is the engine behind the main theorem and can be read as a self-contained statement about closest points on polyhedral cones.
  • For cusps $y^2=x^{2r+1}$, the persistent corner set is $\{j,j+1\}$ with $j=\lceil \alpha(r)\rceil$ for a specific cubic root $\alpha(r)$, giving a closed-form answer for these singularities.
  • The explicit worst 1-PS provides the data needed to build Hesselink-Kempf-Kirwan-Ness stratifications and, potentially, non-reductive GIT quotients of unstable singular curves, which the paper cites as motivation.
  • The paper's computed examples show that the time to persistence can be large (for instance, $N_0=8369$ for the order-9 cusp), so the asymptotic regime is genuinely needed for uniform statements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The tail formula is exactly the least-squares regression line through the last few points, which suggests a variational principle: for large $N$, the dominant cost is matching the flat height-2 tail of $a$, and the optimal graph is the one whose final segment is the regression line; this principle might extend to other one-branch singularities with the same conductor structure.
  • A natural testable extension is to semigroups with two or more gaps above the conductor; persistence should still hold, but the corner set and the constants $\ell,N_0$ may depend on more arithmetic of the semigroup than the examples reveal.
  • The persistence result may be interpretable as an asymptotic stability invariant: one could define the instability type of a singularity by the pair $(\ell, mN+b)$ tail, analogous to how Newton polygons encode singularity data.
  • If the unproved $\mathbb{G}_m$ extension of the Kempf-Morrison Lemma fails, the convex optimization would still compute the best torus one-parameter subgroup; a direct search for a non-torus 1-PS beating the computed ratio in a small example would settle which statement is true.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies the worst destabilizing one-parameter subgroup for the Chow point of a toric rational curve with one unibranch singularity. The authors translate the Hilbert-Mumford ratio maximization into a convex optimization problem: the optimal weight vector is the nearest point on an explicit polyhedral cone W to an explicit vector a (Theorem 2.18). They then use KKT conditions and determinant recurrences to prove a persistence result: for all sufficiently large embedding dimension N, the optimal weights have a fixed initial segment and a linearly descending tail with explicitly given slope and intercept (Main Theorem, Theorem 6.3). Examples for several numerical semigroups and for higher-order cusps are presented, including explicit values of N0.

Significance. If the main theorem is correct, it provides one of the first explicit infinite families of worst destabilizing subgroups in a curve-moduli GIT problem, with direct relevance to Hesselink-Kempf-Kirwan-Ness stratifications and non-reductive GIT constructions. The translation to convex geometry is elegant and parameter-free, and the proof is largely a self-contained pencil-and-paper argument with determinant recurrences in the appendices. The authors are transparent that the proof does not depend on computer calculations, and they provide reproducible code and illustrative tables. The central caveat is that the reduction from GL_{N+1} to the maximal torus is not proved for the infinite group G_m; this is a correctness risk for the main theorem.

major comments (2)
  1. [§2.4, Theorem 2.18] The reduction from arbitrary one-parameter subgroups of GL_{N+1} to the diagonal maximal torus is load-bearing and is not proved. The text states in a parenthetical note that the Kempf-Morrison Lemma [22, Prop. 4.7], proved for finite automorphism groups, 'works for Gm as well,' but no argument is supplied. Since the Gm action has distinct weights on P^N, any 1-PS commuting with it is automatically diagonal, so it suffices to prove, or cite from Kempf [17], the standard theorem that for a connected reductive stabilizer H fixing the point, an optimal 1-PS can be chosen in the centralizer of H. Without this, Theorem 2.18 and all subsequent persistence theorems only describe the best 1-PS inside the torus. Please add a lemma or a precise reference for this extension.
  2. [§2.7, Definition 2.15 and Theorem 2.18] The cone W is defined as the set of convex weight vectors, without imposing nonnegativity, but Lemma 2.16 and the reduction in Theorem 2.18 use nonnegative convex weight vectors. The equality max_{w in W} a·w/||w|| therefore needs justification for vectors with negative coordinates. This can be repaired by observing that, since a_i > 0, replacing w by the coordinatewise maximum max(w,0) preserves convexity and increases the ratio, so the optimum lies in the nonnegative orthant. The paper should either include this argument or define W with the condition w >= 0.
minor comments (4)
  1. [§6.3, proof of Theorem 6.3(i)] The verification that the constructed vector ~x satisfies the KKT equation for rows ℓ+1 through N+1 is compressed to 'we substitute the formulas and verify these identities of rational functions.' Given the intricacy of the formulas for ~x_i, it would be helpful to display the key substitutions or provide a short computer-algebra record so the identities can be checked independently.
  2. [§2.7, Theorem 2.18] Since the optimization in Theorem 2.18 is over real vectors w in W, the paper should state explicitly that a rational optimal direction can be rescaled to obtain an integer-weight 1-PS; the cone and data are rational, so this is immediate but should be said.
  3. [Appendix B] There are a few typographical slips, such as 'Propositon 7.4' in the appendix heading and 'along tis bottom row' in Appendix A; these should be corrected in the final version.
  4. [Remark after Lemma 4.1] The statement that a computer search over numerical semigroups of genus g <= 14 supports a conjecture should include the exact semigroup range and a link to the code used, so that the claim is reproducible.

Circularity Check

1 steps flagged · score 4.0 of 10

Torus reduction rests on an unproved extension of a self-cited Kempf–Morrison lemma; the convex-geometry persistence proof is otherwise self-contained.

  1. self citation load bearing [Section 2.4 (reduction to the maximal torus, before Theorem 2.18)]
    "This allows us to apply the Kempf-Morrison Lemma [22, Prop. 4.7] and conclude that if CΓ is unstable for the GL_{N+1}, then a worst 1-PS will appear in the maximal torus diagonalizing the Gm-action. (Note: the statement in the cited work is for a finite automorphism group, but the proof works for Gm as well.)"

    The derivation of the main theorem begins by restricting GL_{N+1} to the diagonal maximal torus. The sole justification is [22, Prop. 4.7], a paper coauthored by Swinarski, whose stated hypothesis is a finite automorphism group; the Gm case is added in a parenthetical without proof. Theorem 2.18 then identifies the proximum on the torus cone W with a worst 1-PS, and the Persistence theorem is proved only for this proximum. If the extension fails, the computed object is the best torus 1-PS rather than the true worst 1-PS. This is load-bearing self-citation because the reduction is not derived in the present paper and the cited source does not contain the needed statement.

full rationale

Apart from the torus-reduction step, the paper's derivation is self-contained and non-circular. The convex-geometry translation (Theorem 2.18) is a direct application of the lower-convex-hull formula and the trapezium computation; the KKT analysis treats the cone W and vector a as fixed inputs derived from the semigroup Γ; and the persistence theorem is proven by determinant and polynomial arguments, not by fitting the output. The 'least squares regression line' in Lemma 6.2 and Theorem 6.3 is not a fitted parameter renamed as a prediction: it is the exact minimizer of the tail of the quadratic objective, with slope and intercept explicit functions of N and ℓ. The self-citation [22] is real evidence for the finite-automorphism-group case, but the extension to the infinite group Gm is asserted, not proved. This is a load-bearing gap in the reduction from GL_{N+1} to the torus; however, it does not make the persistence conclusion equivalent to an input by construction. Hence the score is 4 rather than 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof relies on standard GIT, toric geometry, and convex optimization background. The only non-standard premise is the unproved extension of the Kempf-Morrison Lemma to Gm. There are no free parameters fitted to data and no invented entities.

assumptions (5)
  • standard math Existence and uniqueness of the worst 1-PS for unstable points (Kempf and Mumford).
    Used in Definition 2.2 and Theorem 2.3; standard GIT background.
  • domain assumption Kempf-Morrison Lemma, and its asserted extension to the infinite group Gm acting with distinct weights.
    Section 2.4 relies on this to restrict the search to the maximal torus; the extension is stated without proof.
  • standard math Identification of the Chow polytope with the secondary polytope, and the volume formula for the Hilbert-Mumford function (Lemma 2.10).
    Cited from Gelfand-Kapranov-Zelevinsky [6]; used in Proposition 2.11.
  • standard math Karush-Kuhn-Tucker conditions are sufficient and necessary for the strongly convex quadratic nearest-point problem on the cone W.
    Section 3.1; standard convex optimization background.
  • domain assumption The numerical semigroup Γ has finite complement and conductor; the parametrization CΓ has the stated Gm-action with distinct weights.
    Definitions 2.5-2.7; true for the curves considered.

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Pith. "Pith review of The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity." pith.science (2026). https://pith.science/paper/YDGXXLVQ

@misc{pith2026250200458,
  author       = {Pith},
  title        = {Pith review of: The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDGXXLVQ}},
  note         = {Machine review of arXiv:2502.00458}
}
abstract

Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a ``worst'' destabilizing 1-parameter subgroup $\lambda^{*}$. It is natural to ask: what are the worst 1-PS for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric rational curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples.

Figures

Figures reproduced from arXiv: 2502.00458 by the authors.

Figure 1
Figure 1. The vector a and the optimal weights w for the Simplified Problem for Γ = h2, 3i and N = 10 3. The KKT matrix equation and its solutions To solve the optimisation problem described in the previous section, we will use the Karush-Kuhn-Tucker (KKT) conditions in nonlinear optimisation to study the closest point on the span of each face of W to the vector a. Recall that we write cond(Γ) for the conductor of Γ, and c. i… view at source ↗

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