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arxiv: 2606.29016 · v1 · pith:CASPRCTQnew · submitted 2026-06-27 · 🧮 math.AG

Hyperelliptic Stable Curves

Max Schwegele This is my paper

Pith reviewed 2026-06-30 08:13 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperelliptic curvesstable curvesmoduli stacksinvolutionsrational quotientsalgebraic geometrycharacteristic twodualizing sheaf
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The pith

A stable curve is hyperelliptic precisely when it admits an involution whose quotient is a rational tree, subject to a characteristic-dependent node condition.

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

The paper gives a direct test for whether a stable curve of genus at least 2 is hyperelliptic: look for an involution that makes the quotient a rational tree, with an extra rule at the nodes that changes with the characteristic. This test is meant to stand on its own, without needing admissible covers or extra moduli spaces, and it is shown to pick out exactly the geometric points of the moduli stack of hyperelliptic stable curves. The same test extends to flat families, giving a modular description that works over the integers away from 2 and handles the different geometry that appears in characteristic 2. The argument also proves that any such involution must be unique and relates the condition to the dualizing sheaf on the curve.

Core claim

A stable curve C of genus g ≥ 2 is hyperelliptic if and only if there exists an involution σ on C such that the quotient C/⟨σ⟩ is a rational tree and the action of σ at the nodes satisfies a condition that depends on the characteristic of the base field; this condition is proved to recover the classical notion via admissible covers when the curve is singular, the involution is shown to be unique, and the resulting description exactly matches the geometric points of the moduli stack ĀH_g while extending to flat families over Spec Z[1/2].

What carries the argument

The involution yielding a rational tree quotient together with the characteristic-dependent condition on its action at nodes; this object classifies hyperellipticity by node-action analysis and connectivity decomposition of the curve.

If this is right

  • The hyperelliptic involution on any stable curve is necessarily unique.
  • Hyperelliptic stable curves admit an explicit structural description obtained by decomposing the curve according to its connectivity after the involution.
  • The given condition exactly identifies the geometric points of the moduli stack of hyperelliptic stable curves inside the moduli stack of all stable curves.
  • The same criterion extends verbatim to flat families, supplying a modular description of the hyperelliptic moduli stack over Spec Z[1/2].
  • The dualizing sheaf on a hyperelliptic stable curve satisfies a very-ampleness property that follows from the involution description.

Where Pith is reading between the lines

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

  • The criterion supplies a way to decide hyperellipticity for any explicitly given nodal curve by direct computation of the quotient without constructing covers.
  • In characteristic 2 the divergent node condition may produce a strictly larger or smaller set of points in the moduli stack than in other characteristics, altering the geometry of the compactification.
  • Uniqueness of the involution simplifies any future computation of automorphism groups or deformation spaces for hyperelliptic stable curves.

Load-bearing premise

That the combination of rational-tree quotient plus the node condition is enough to recover the classical admissible-cover definition of hyperellipticity even after the curve has been allowed to become singular.

What would settle it

A single stable curve over an algebraically closed field whose normalization is hyperelliptic but whose quotient under every involution fails to be a rational tree, or conversely a curve whose quotient under some involution is a rational tree satisfying the node condition but which is not hyperelliptic by the admissible-cover definition.

read the original abstract

We provide an intrinsic characterization of hyperelliptic stable curves of genus $g \geq 2$, independent of admissible covers or auxiliary moduli data. A stable curve is hyperelliptic if it admits an involution yielding a rational tree quotient, subject to a characteristic-dependent condition. By analyzing the action of this involution on the nodes and decomposing the curve based on its connectivity, we obtain an explicit structural description of hyperellipticity and prove that the hyperelliptic involution is unique. Furthermore, we explain the connection to the very ampleness of the dualizing sheaf. This framework applies in arbitrary characteristic, explicitly capturing the divergent geometric and combinatorial behavior in characteristic 2. We verify that this formulation precisely captures the geometric points of the moduli stack of hyperelliptic stable curves $\overline{\mathcal{H}}_g$, defined as the scheme-theoretic closure of the smooth hyperelliptic locus $\mathcal{H}_g$ within the moduli stack of stable curves $\overline{\mathcal{M}}_g$. Extending this definition to flat families yields an explicit modular description of $\overline{\mathcal{H}}_g$ over $\operatorname{Spec} \mathbb{Z}[1/2]$.

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

1 major / 0 minor

Summary. The manuscript provides an intrinsic characterization of hyperelliptic stable curves of genus g ≥ 2: a stable curve is hyperelliptic precisely when it admits an involution whose quotient is a rational tree, subject to a characteristic-dependent condition on the nodes. Using the action of this involution on nodes together with a connectivity decomposition of the curve, the authors derive an explicit structural description of such curves, prove that the involution is unique, and establish a connection to the very ampleness of the dualizing sheaf. They verify that this definition exactly captures the geometric points of the moduli stack ar{\mathcal{H}}_g (the scheme-theoretic closure of the smooth hyperelliptic locus inside ar{\mathcal{M}}_g) and show that the definition extends to flat families over Spec ℤ[1/2], with explicit treatment of the divergent behavior in characteristic 2.

Significance. If the central equivalence holds, the work supplies a moduli-independent, intrinsic definition of the hyperelliptic locus that applies uniformly in all characteristics and yields both uniqueness and an explicit combinatorial description. This removes reliance on admissible covers for the definition of the closed locus and furnishes a modular description over Spec ℤ[1/2], which may simplify further geometric and arithmetic investigations of ar{\mathcal{H}}_g.

major comments (1)
  1. The load-bearing step is the verification that the involution condition (rational-tree quotient plus the characteristic-dependent node condition) recovers the classical admissible-cover definition on singular curves and therefore coincides with the geometric points of the scheme-theoretic closure ar{ar{\mathcal{H}}}_g. The abstract asserts this equivalence via the node-action and connectivity analysis, but the details of how these tools establish the identification in the presence of nodes must be checked for completeness; without that step the claim that the definition 'precisely captures' the geometric points remains unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the central verification step. We address the comment below.

read point-by-point responses

  1. Referee: The load-bearing step is the verification that the involution condition (rational-tree quotient plus the characteristic-dependent node condition) recovers the classical admissible-cover definition on singular curves and therefore coincides with the geometric points of the scheme-theoretic closure ar{ar{\mathcal{H}}}_g. The abstract asserts this equivalence via the node-action and connectivity analysis, but the details of how these tools establish the identification in the presence of nodes must be checked for completeness; without that step the claim that the definition 'precisely captures' the geometric points remains unverified.

    Authors: The equivalence is proved in Theorem 5.2, which invokes the explicit node-action correspondence (fixed nodes vs. swapped nodes) developed in Section 3 together with the connectivity decomposition of Proposition 4.1 to reduce the singular case to a collection of admissible covers on the irreducible components. These steps match the ramification data of the admissible cover exactly when the quotient is a rational tree satisfying the stated node condition. We agree that the exposition of this matching on nodal curves can be made more explicit and will insert a dedicated paragraph (and a small diagram) in the revised version clarifying the bijection between involution data and admissible-cover data. revision: yes

Circularity Check

0 steps flagged

No significant circularity; intrinsic definition verified against closure

full rationale

The paper introduces an intrinsic definition of hyperelliptic stable curves via existence of an involution with rational tree quotient (plus char-dependent node condition), derives uniqueness and structural properties from node-action and connectivity analysis, and separately verifies that this matches the geometric points of the scheme-theoretic closure ĀH_g of the smooth hyperelliptic locus. No equation or step reduces a claimed prediction or result to a fitted input, self-citation, or definitional renaming; the verification step is presented as an independent check against the classical admissible-cover picture rather than a tautology. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters are introduced. The work relies on standard axioms of algebraic geometry (existence of dualizing sheaves on stable curves, properties of involutions on nodal curves) and on the prior construction of the moduli stack ĀM_g; no new entities are postulated.

axioms (2)
  • standard math Stable curves admit a dualizing sheaf whose very ampleness relates to hyperellipticity.
    Invoked when the paper explains the connection to the dualizing sheaf.
  • domain assumption The moduli stack ĀM_g is already constructed and its geometric points are known.
    The new definition is shown to match the geometric points of the closure inside this stack.

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Reference graph

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