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Semistable Reduction of Plane Quartics

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Pith's one-line read This paper proves that a plane quartic over a discretely valued field admits a GIT-stable plane model precisely when its stable reduction is non-hyperelliptic, and describes the domination map: it contracts the 1-tails of the stable fiber t

desk verdict A genuinely new bridge between GIT-stable plane quartics and stable reduction, with a real but likely fillable gap in the residue-reciprocity input that the author himself flags. read the letter →

arxiv 2511.15858 v2 pith:EVWHWA4D submitted 2025-11-19 math.AG math.NT

Max Schwegele This is my paper · ORCID
classification math.AGmath.NT MSC 14H1014H4514L2414H5014G20
keywords stablereductionplanequarticsGITstabilitygenus3curveshyperellipticdualizingsheaf1-tailsresiduereciprocity
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

For a smooth plane quartic (equivalently, a non-hyperelliptic genus-3 curve), the paper establishes that two a priori unrelated notions of stability coincide: geometric Deligne–Mumford stability of the reduction and GIT stability of a plane model over the valuation ring. The equivalence is constructive — when the stable reduction is non-hyperelliptic, the stable model maps onto a GIT-stable plane model by a morphism whose special fiber contracts each elliptic or pig tail to a cusp. Conversely, existence of a GIT-stable plane model forces the stable reduction to be non-hyperelliptic, and the stable model is the minimal semistable model dominating it. This matters because GIT-stable models are computable, giving a route to compute stable reductions in residue characteristics 2 and 3 where classical admissible reduction is unavailable. The central technical step is a twist of the dualizing sheaf along the tails, whose sections are shown to be exactly the differentials with vanishing residues at the attachment points.

What carries the argument

The load-bearing object is a twisted line bundle L = ω_{X/O}(D) on the stable model, where D is the sum of the thickness-weighted tail components. Restricting to the special fibre, the 3-dimensional space V of lifted sections is shown (via residue reciprocity along flags on the model) to coincide with the subspace W of differentials on the core with vanishing residues at the attachment points. This identification turns the twisted linear series into the base-point-free canonical linear series of a cuspidal curve C' formed by contracting the attachment points to A2-singularities, producing the GIT-stable plane quartic; the tails are contracted because they contribute only constant sections.

What would settle it

Take a smooth plane quartic over a 2-adic field whose reduction type is non-hyperelliptic (e.g., an irreducible core of genus 2 with a node, type 2n in the paper's list), compute the subspace of H^0(core, ω(2x1+...+2xr)) cut out by vanishing residues at the attachment points, and check whether its dimension is 3 and its linear series base-point-free. A counterexample would be a single type-2n or BRAID curve where this dimension drops below 3, forcing base points and contradicting the identification of the two subspaces.

Watch

Extended reading notes

Core claim

The central discovery is an exact dictionary between abstract stable reduction and geometric invariant theory for plane quartics. Theorem 1.15 states: the stable reduction is non-hyperelliptic if and only if a GIT-stable plane model exists. Under this condition the stable model is the unique minimal semistable model dominating the GIT-stable plane model, and the domination morphism's special fibre contracts the 1-tails (arithmetic-genus-one components attached at a single node) to cusps while embedding the core. In the hyperelliptic case no GIT-stable model exists and the same twisted linear series instead collapses the core two-to-one onto a double conic. The proof identifies the three-dime

Load-bearing premise

The equivalence rests on assuming the model's residue reciprocity laws hold in every residue characteristic, though the cited theory was developed for characteristic 0; if these laws fail in characteristic 2 or 3, the whole bridge between GIT-stable and stable models collapses.

Editorial extensions

If this is right

  • In every residue characteristic, a non-hyperelliptic stable reduction of genus 3 can be realized as the minimal resolution of the cuspidal singularities of a GIT-stable plane quartic.
  • Computing a GIT-stable plane model becomes a viable route to the stable model, bypassing admissible-cover techniques that break down for p = 2 and p = 3.
  • The hyperelliptic case is characterized by degeneration of the model to a double conic, giving a geometric criterion to detect hyperellipticity of the reduction from plane equations alone.
  • The classification of the 42 genus-3 reduction types reduces the hyperelliptic/non-hyperelliptic question to core geometry: 2-separable cores are always hyperelliptic, while 2-inseparable cores depend on Weierstrass points or cross-ratios, with characteristic dependence only in the four rational-core types.
  • The canonical-map criterion for very strongly connected curves becomes the uniform proof that 2-inseparable non-hyperelliptic stable curves embed by their canonical system, replacing case-by-case Riemann–Roch checks.

Reading between the lines

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

  • If the reciprocity hypotheses fail in small characteristic, the equivalence could break precisely in the motivating cases p = 2 and p = 3; a direct residue calculation on the local model OK[[u,v]]/(uv − π^d) in equal characteristic 2 would be the sharpest test.
  • The mechanism suggests a general template: twisting the dualizing sheaf by separating components of a stable reduction and matching residue-vanishing subspaces may produce GIT-stable models for other canonically embedded curves.
  • A practical algorithm for stable reduction in small characteristic would follow: compute a GIT-semistable plane quartic, detect GIT-stability by singularity type, then resolve the resulting cusps; the paper states this algorithmic extraction as future work.
  • The dichotomy — non-hyperelliptic core maps to a curve, hyperelliptic core maps to a double conic — gives a purely plane-model way to test for hyperellipticity of the reduction without computing the stable model.
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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

3 major / 5 minor

Summary. The paper proves a sharp bridge between Deligne–Mumford/stable reduction and GIT-stable plane models for smooth plane quartics over a complete discretely valued field with algebraically closed residue field. The main theorem (Theorem 1.15) asserts: the stable reduction is non-hyperelliptic if and only if a GIT-stable plane model exists; when this holds, the stable model is the unique minimal semistable model dominating the GIT-stable model, and the domination morphism contracts the 1-tails to cusps and is an immersion elsewhere. Chapter 2 gives a detailed classification of hyperelliptic stable curves of genus 3, including characteristic-2 phenomena. Chapter 3 constructs the contraction map from the stable model by twisting the dualizing sheaf by the 1-tails, identifies the resulting linear series with the residue-vanishing subspace, and uses Catanese's theorems to prove base-point-freeness and the GIT-stability of the image. The converse direction is announced and sketched via the 'stable hull' of a GIT-stable model.

Significance. If the main theorem is fully established, it gives a genuinely useful computational framework for stable reduction of plane quartics, particularly in residue characteristics 2 and 3, where admissible-cover methods are difficult or inefficient. The conceptual identification of GIT-stability of plane quartic models with non-hyperellipticity of stable reduction is natural and important. The paper also provides a clear and fairly complete classification of hyperelliptic stable curves of genus 3, with careful attention to characteristic 2. Strengths include the use of external benchmarks (Catanese's canonical-map theorems, Deligne–Mumford/Liu stable reduction, and the classical GIT classification of quartic singularities), the explicit construction of the twisted line bundle, and the geometric description of the contraction morphism.

major comments (3)
  1. [§3.3, Prop. 3.45 and Remark 3.18; Theorem 3.33] The proof of the implication (i)=>(ii) depends on the identification V≅W in Proposition 3.45, and specifically on the inclusion V⊆W. That inclusion is obtained by combining the reciprocity law around a point and the reciprocity law along a vertical curve (Theorem 3.33), both cited from Morrow's theory. Morrow's results are stated for characteristic-0 local fields with finite residue field, while the paper's motivating cases are p=2 and p=3, in both mixed and equal characteristic. The author explicitly declines to develop the required generalization in Remark 3.18 ('would extend beyond the scope of this work'), and Theorem 3.31 is likewise only 'expected to hold in greater generality'. This is a load-bearing point: if the reciprocity laws fail or require additional hypotheses in the target residue characteristics, then the proof of V⊆W collapses, Lemma 3.46(iii) cannot be invoked, and the
  2. [§3.3, Proof of Theorem 3.43 and Lemma 3.48] The reduction to Catanese's theorem passes through the curve C′ obtained by gluing cusps at the attachment points of the 1-tails. The proof of Lemma 3.48, especially the direction (ii)=>(iv), contains local computations in the completed rings of the normalization, including the wild case chark=2. These local computations are delicate and are only presented for the completion of a smooth point of Cc. The argument should verify that the factorization of the degree-two map through the cuspidal curve is independent of the choice of local uniformizer and that the invariant-ring inclusions proved in the completed local rings imply the claimed global factorization; as written, the step from x to a global morphism is somewhat compressed. This is not independent of the previous comment, but it is part of the same load-bearing chain and deserves a fuller write-up.
  3. [§1.2, Definition 1.14] The paper uses the classical GIT classification of plane quartics as a working definition, with singularity types A1 and A2. In positive characteristic, especially p=2, the classification of singularities and the relationship between GIT stability and these normal forms require care. The paper notes that the normal forms for A1 and A2 hold in any characteristic and cites [GK90] for An in characteristic 2, but the rest of the classification is quoted from [Art09], [MFK94], and [Hui79], whose characteristic assumptions are not fully discussed. Since the main theorem is about arbitrary residue characteristic, the statement of Definition 1.14 should either be explicitly treated as the definition used in this paper (in which case the later appeal to 'GIT-stable' is a theorem about this class), or the characteristic dependence of the GIT classification should be clarified. This affects the pre
minor comments (5)
  1. [Notation throughout] The stable model is denoted X and its special fiber is also denoted X. This is a common but potentially confusing convention; a different symbol for the special fiber, perhaps X_s, would improve readability in Chapter 3 where both appear in the same arguments.
  2. [§3.1, Definition 3.19] The definition of a flag as a pair (x,y) with y a codimension-one point specializing to x is clear, but the terminology 'flag' is not standard in all references; a brief reminder of the geometric meaning (horizontal vs. vertical curve through x) would help the reader who is not already familiar with Morrow's setup.
  3. [§3.2, Lemma 3.35] The statement and proof of Lemma 3.35 are useful. The proof of part (b) invokes [EGA III1, Proposition 4.6.7(i)]; a more elementary reference or a one-line explanation of why the closed-immersion property spreads from the special fiber over a DVR would make the argument more self-contained.
  4. [§3.3, Lemma 3.44] The proof of linear independence of the first r−1 residue conditions would be clearer if the two displayed dimensions were explicitly justified by Riemann–Roch: the first is h^0(X_c, ω( x_j+2x_r)) = (3−r)+ (1+2) −1 = 5−r, and the second is (3−r)+2−1 = 4−r. Adding this one line would remove unnecessary computation for the reader.
  5. [§2.3, Proposition 2.38] The table of 42 combinatorial types is very useful, but the use of the shorthand '0n' for a self-intersection on a genus-0 component and the distinction between arithmetic and geometric genus in the naming convention could be stated even more prominently. The current Remark 2.37 helps, but a small glossary near the table would improve accessibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivation is self-contained against external benchmark results; the residue-theory generality question is a correctness gap, not an input/output identity.

full rationale

The paper's central claim is the equivalence (i) stable reduction non-hyperelliptic ⇔ (ii) GIT-stable plane model exists, plus the description of the stable model as the minimal semistable model dominating the GIT-stable model. The proof builds a morphism from the stable model using the twisted line bundle L = ω_{X/O_K}(D), with D supported on the 1-tails. The key base-point-freeness step is Proposition 3.45, which identifies the 3-dimensional subspace V (images of sections of L on the special fiber) with the residue-vanishing subspace W. This identification is proved by residue and reciprocity arguments, not by defining V in terms of the target plane model or the GIT-stable quartic. V and W are independent objects: V comes from lifting global sections from the model, while W is defined by vanishing residues at attachment points; Proposition 3.45 proves V ≅ W rather than assuming it. The classification of hyperelliptic stable genus-3 curves is established using independent external frameworks: Liu's treatment of smooth curves, ACG11/Mau03 for admissible covers and hyperelliptic involutions, and Catanese's theorems on canonical maps. Corollary 2.60, which is used to embed 2-inseparable non-hyperelliptic stable curves, is a derived consequence of Catanese's Theorem G, not a restatement of the paper's target. The citations to [vBDLLG25] are used for comparison, for the r=0 special case, and as motivation; the relevant result (Proposition 3.41) is proved independently via Corollary 2.60. The paper does invoke Morrow's reciprocity laws (Theorems 3.33) and notes that a full generalization to arbitrary residue characteristic is not developed (Remark 3.18). This is a real technical gap that could undermine the proof if the generalization fails, but it is not circularity: it is an appeal to an external result whose hypotheses may not cover the intended setting. No fitted parameter is relabeled as a prediction, no ansatz is smuggled in via self-citation, and the claimed uniqueness properties are backed by Deligne–Mumford–Liu stable reduction theory and the GIT classification of quartic singularities, not by a self-referential chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on four kinds of input: classical stable-model theory (DM69, Liu02), Catanese's external theorems on canonical maps of Gorenstein curves, the external identification of geometric vs moduli-theoretic hyperellipticity (Maugeais/Yamada), and — the only item carried ad hoc — an unproved extension of Morrow's arithmetic residue theory to arbitrary residue characteristic. No fitted constants appear; the only 'choice' in the construction, the twisting divisor D = Σ d_i X_i, is determined by the model's node thicknesses rather than chosen to force the conclusion.

assumptions (6)
  • standard math Deligne-Mumford stable reduction theorem; existence and normality of the unique minimal stable model over a complete DVR.
    Theorem 1.5 and Proposition 1.6; background framework from [DM69] and [Liu02]. The whole setup in Setup 1.1 rests on this.
  • standard math Catanese's theorems on pluricanonical maps, canonical maps of inseparable Gorenstein curves, and the classification of very strongly connected quasi-hyperelliptic curves ([Cat82] Theorems D, E, F, G).
    Used in Corollary 2.60, Proposition 2.55, and Lemma 3.48. Remark 2.59 correctly notes one of Catanese's auxiliary conjectures is unproven in general but only the proven node case is used.
  • standard math The geometric definition of a hyperelliptic stable curve (involution with rational-tree quotient) agrees with the moduli-theoretic closure of the smooth hyperelliptic locus in all characteristics.
    Section 2.2 opening; anchored to [ACG11] (k=C), [Mau03, Théorème 5.4] and [Yam04, Appendix A.2] (positive characteristic incl. p=2).
  • ad hoc to paper Morrow's arithmetic residue theory and reciprocity laws on the model extend from characteristic-0 local fields with finite residue field to arbitrary residue characteristic (equal and mixed characteristic).
    Remark 3.18: 'A detailed development of this generalization would extend beyond the scope of this work, as we only need the theory for a single result (Proposition 3.45).' Theorem 3.31 similarly 'expected to hold in greater generality'. Proposition 3.45 and hence the main theorem depend on this.
  • domain assumption The GIT classification of plane quartics over the residue field is as stated in Definition 1.14, including the singularity normal forms in small characteristic.
    Definition 1.14 with references [Art09], [MFK94]; singularity classification attributed to Hui [Hui79]; the paper cites [vBDLLG25, Remark 2.3] for the p>7 caveat on the classification, with [GK90] for the characteristic-2 normal forms.
  • domain assumption The residue field k is algebraically closed (weakened to perfect).
    Setup 1.1 and Remark 1.16: 'the weaker assumption that k is perfect would suffice' via the arguments in [Liu02, Chapter 10].

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Pith. "Pith review of Semistable Reduction of Plane Quartics." pith.science (2026). https://pith.science/paper/EVWHWA4D

@misc{pith2026251115858,
  author       = {Pith},
  title        = {Pith review of: Semistable Reduction of Plane Quartics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVWHWA4D}},
  note         = {Machine review of arXiv:2511.15858}
}
abstract

The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite field extension. Computing this model is a central problem in arithmetic geometry. For non-hyperelliptic genus $3$ curves, which are canonically embedded as plane quartics, methods like admissible reduction become challenging in small residue characteristics. This thesis establishes a precise connection between the abstractly defined stable model and computationally accessible GIT-stable plane models. We prove that a GIT-stable plane model of a smooth plane quartic exists if and only if its stable reduction is non-hyperelliptic. When this condition holds, we show that the stable model is the unique minimal semistable model that dominates the GIT-stable model. The corresponding domination morphism is geometrically explicit: it contracts the $1$-tails of the stable reduction to cusps on the special fiber of the GIT-stable model and is an immersion elsewhere. This result provides a geometric framework for computing the stable model by first finding a GIT-stable model and then resolving its cuspidal singularities.

Figures

Figures reproduced from arXiv: 2511.15858 by the authors.

Figure 1
Figure 1. The 7 combinatorial types of stable curves of genus 2. seven configurations can be grouped into three irreducible types, three types formed by two curves joined at a single node, and a final type, a binary curve, in which two smooth rational curves meet at three distinct points. In the figure, the geometric genus of each component (0, 1, or 2) is indicated by line thickness, from thinnest to thickest. This classific… view at source ↗
Figure 2
Figure 2. Examples of the properties in Definition [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. The decomposition process of a curve of type [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗

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  1. Semistable reduction of smooth quartics

    math.AG 2026-06 conditional novelty 4.0 of 10

    For smooth plane quartics, non-hyperelliptic stable reduction is equivalent to the existence of a unique GIT-stable plane model, and the stable model is obtained by cusp resolution.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.