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arxiv: 2507.01232 · v3 · submitted 2025-07-01 · 🧮 math.AG

Torus actions, weighted blow-ups, and desingularization of plane curves

Dan Abramovich , Ming Hao Quek , Bernd Schober This is my paper

Pith reviewed 2026-05-19 05:56 UTC · model grok-4.3

classification 🧮 math.AG MSC 14E15
keywords resolution of singularitiesweighted blow-upstorus actionsplane curvesembedded resolutiondesingularizationalgebraic surfaces
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The pith

Weighted blow-ups at torus-equivariant centers resolve singularities of plane curves over any field.

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

The paper constructs an embedded resolution of singularities for a singular hypersurface inside a regular two-dimensional scheme by a sequence of weighted blow-ups. It obtains an inductive proof by selecting centers that respect torus actions, which handles the higher-dimensional tangent spaces that appear during the process. The argument works without assuming the base field is perfect and avoids the multi-weighted blow-ups required in earlier constructions. A reader would care because the method supplies an explicit, field-independent procedure for desingularizing curves that can be carried out directly on the ambient surface.

Core claim

Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by weighted blow-ups. We deduce an inductive argument, despite the fact that higher dimensional tangent spaces arise, by taking torus actions and equivariant centers into account. In addition, we do not have to restrict to perfect base fields.

What carries the argument

Torus actions and equivariant centers that guide the choice of weighted blow-up centers, enabling induction to proceed through steps where tangent spaces increase in dimension.

Load-bearing premise

Suitable torus actions exist on the ambient scheme so that equivariant centers can always be chosen to continue the inductive resolution.

What would settle it

A concrete singular curve in a regular surface over a field for which no finite sequence of weighted blow-ups at torus-equivariant centers produces a regular total transform.

read the original abstract

Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by weighted blow-ups. This differs from our earlier work which required multi-weighted blow-ups. We deduce an inductive argument, despite the fact that higher dimensional tangent spaces arise, by taking torus actions and equivariant centers into account. In addition, we do not have to restrict to perfect base fields.

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

2 major / 2 minor

Summary. The paper constructs an embedded resolution of singularities for a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, achieved via iterated weighted blow-ups. It develops an inductive argument that accommodates higher-dimensional tangent spaces at singular points by incorporating torus actions and equivariant centers, and it extends the result to imperfect base fields without requiring multi-weighted blow-ups as in prior work.

Significance. If the construction holds, the result supplies a concrete algorithm for resolving plane curve singularities over arbitrary fields using only weighted blow-ups, which may simplify existing resolution procedures and enable new applications in computational algebraic geometry over non-perfect fields. The explicit use of torus actions to manage induction represents a technical advance in handling non-standard tangent spaces.

major comments (2)
  1. [§3] §3 (Inductive step): The argument that a suitable torus action always exists making both the hypersurface and the chosen weighted center equivariant is load-bearing for the induction. The manuscript must supply an explicit construction or existence proof for this action when the base field is imperfect and the tangent space dimension at the singular point exceeds 1; without it, the reduction to lower multiplicity or simpler singularities may fail for some hypersurfaces.
  2. [§4.2] §4.2, Definition of weighted center: The multiplicity and weight choices for the center must be shown to be compatible with the torus action in all cases, including when the scheme is not smooth over the base field. The current sketch leaves open whether the equivariance condition can always be satisfied simultaneously with the multiplicity drop required for induction.
minor comments (2)
  1. [§2] Notation for the torus action and the weighted blow-up morphism should be introduced earlier and used consistently throughout the inductive argument to improve readability.
  2. [Introduction] The comparison with the authors' earlier work on multi-weighted blow-ups would benefit from a short table or explicit list of differences in the hypotheses and conclusions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance. We address the two major comments point by point below, clarifying the relevant arguments and indicating the revisions we will make to improve explicitness.

read point-by-point responses

  1. Referee: [§3] §3 (Inductive step): The argument that a suitable torus action always exists making both the hypersurface and the chosen weighted center equivariant is load-bearing for the induction. The manuscript must supply an explicit construction or existence proof for this action when the base field is imperfect and the tangent space dimension at the singular point exceeds 1; without it, the reduction to lower multiplicity or simpler singularities may fail for some hypersurfaces.

    Authors: In §3 we construct the required torus action explicitly by selecting a regular system of parameters whose initial forms diagonalize the lowest-degree homogeneous component of the hypersurface equation with respect to the chosen weights. When the tangent space has dimension greater than 1, the construction proceeds by lifting a basis of the cotangent space that makes the initial form quasi-homogeneous; this choice depends only on the associated graded ring and does not invoke separability of the residue field extension. Consequently the same argument applies verbatim over imperfect fields. To address the referee’s request for greater explicitness we will insert a new lemma (Lemma 3.5) that isolates this coordinate choice and verifies equivariance of both the hypersurface and the center. revision: yes

  2. Referee: [§4.2] §4.2, Definition of weighted center: The multiplicity and weight choices for the center must be shown to be compatible with the torus action in all cases, including when the scheme is not smooth over the base field. The current sketch leaves open whether the equivariance condition can always be satisfied simultaneously with the multiplicity drop required for induction.

    Authors: Section 4.2 defines the weights via the orders of vanishing of the chosen parameters in the regular local ring of the ambient scheme. Because the ambient scheme is regular, these orders are well-defined even when the scheme is not smooth over an imperfect base field. The torus action is chosen precisely so that each parameter is an eigenvector; the resulting weighted center therefore remains invariant, and the initial form of the hypersurface acquires positive weighted degree, guaranteeing the multiplicity drop. We will add a short proposition after Definition 4.2 that records this simultaneous verification in a single statement. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction of resolution via weighted blow-ups and torus actions.

full rationale

The paper presents an explicit construction of an embedded resolution of singularities for a singular hypersurface in a regular 2-dimensional scheme by iterated weighted blow-ups. It contrasts this with prior work requiring multi-weighted blow-ups and invokes torus actions plus equivariant centers solely to enable the inductive step when higher-dimensional tangent spaces appear. No step reduces a claimed result to a fitted parameter, self-defined quantity, or unverified self-citation chain; the existence of suitable torus actions is treated as part of the construction rather than an output derived from the resolution itself. The derivation is therefore self-contained as a mathematical algorithm in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts from algebraic geometry about regular schemes, blow-ups, and torus actions. No free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Regular 2-dimensional schemes essentially of finite type over a field admit weighted blow-ups that can be chosen equivariantly with respect to torus actions.
    Invoked to construct the resolution and to run the inductive argument.

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Forward citations

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper

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