{"id":"27a6ddcb-34e5-4f7d-a9d2-41953b23122a","arxiv_id":"2606.13863","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper develops a method to compute the stable reduction of smooth plane quartic curves over p-adic fields, including the hard case of residue characteristic p=2. It shows that a quartic has a GIT-stable plane model exactly when its stable reduction is non-hyperelliptic, and that the stable model is then obtained by resolving the cusps of that model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Forward direction of Theorem 3.1 relies on Catanese [6] in char 2 and on an unproved transfer of '2-inseparable' to 'very strongly connected' for the cuspidal curve C'; this is deferred to [29] and not verified.","rationale":"I read the paper in good faith: Theorem 3.1 is the central claim, and the surrounding strategy—GIT-stable plane models plus explicit cusp resolution—is concrete and supported by reproducible code and random experiments. The argument for the forward direction, however, passes through a point where the proof relies on the positive-characteristic validity of Catanese's theorems and on an unproved identification of two connectivity notions. This is not an internal inconsistency, but it is a load-bearing external dependence; Catanese's own 'honestly hyperelliptic' distinction shows that characteristic 2 is genuinely delicate here. The paper itself flags that Sections 1 and 3 are based on [29] and that the detailed proof is deferred there, so the concern is not manufactured. The reader's conditional verdict already captures this risk; my stress test does not move it to accept or reject, because the experiments and the explicit local resolution provide meaningful evidence in favor of the theorem, while the Catanese step prevents full acceptance. I mark agreement as 'partial' because the reader emphasized the Catanese hypotheses, and I add the specific missing link: the transfer of 2-inseparability from the nodal core to the cuspidal curve C'.","tokens_in":24561,"tokens_out":19949,"duration_ms":207372,"concrete_test":"Obtain the original statements of Catanese [6, Theorems D, F, G] and check their characteristic hypotheses for Gorenstein curves with A1/A2 singularities; then verify directly from Catanese's definition that the curve C' arising from a 2-inseparable core via Lemma 3.7 is very strongly connected. As a computational cross-check, take a characteristic-2 example from §4 (e.g., the 0nee Ciani quartic), compute the canonical map of C' over k = F_2 explicitly, and verify that it is a closed immersion whose image has only A1/A2 singularities. If any computed image has a worse singularity, or if the very-strong-connectivity transfer fails, the forward direction of Theorem 3.1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The forward direction of Theorem 3.1 (§3.2) hinges on the assertion, after Lemma 3.8, that the complete canonical system |ω_C'/k| of the cuspidal Gorenstein curve C' is base-point-free and gives a closed immersion, via Catanese [6, Theorems D, F, G]. The proof states: 'Since C' is a 2-inseparable canonically positive Gorenstein curve, |ω_C'/k| is base-point-free by a theorem of Catanese [6, Theorem D]' and later 'C' is constructed from the 2-inseparable core, so it is very strongly connected [6, Definition 3.21]'. This is the least secure step. The term '2-inseparable' is defined in Definition 1.5 only for semistable (nodal) curves, whereas C' has A2 cusps; the transfer to Catanese's 'very strongly connected' is asserted, not proved. The paper also does not check that Catanese's theorems remain valid for Gorenstein curves with A1/A2 singularities in characteristic 2, where Catanese himself distinguishes 'hyperelliptic' from 'honestly hyperelliptic'. Lemma 3.9 only rules out finite degree-2 maps to P^1; the conclusion that C' is not hyperelliptic in the sense needed for [6, Theorem G] depends on [6, Theorem F]. If that equivalence fails for such C', or if C' is not very strongly connected, then |ω_C'| may fail to be base-point-free or a closed immersion, and the construction would not produce a GIT-stable plane model. Key ingredients (Proposition 3.6's independence, Lemmas 3.7–3.9) are also deferred to [29] and are not independently verified here.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a method for computing stable reduction of smooth plane quartics over discretely valued fields, with emphasis on residue characteristic p=2. It builds on previous work [33] to construct GIT-semistable plane models and then states its main result, Theorem 3.1: assuming X has semistable reduction and stable model X, X admits a unique GIT-stable plane model X0 if and only if the stable special fiber Xs is non-hyperelliptic; in that case X dominates X0 and contracts the 1-tails of Xs to cusps of X0,s, being an isomorphism elsewhere. The proof combines residue theory on arithmetic surfaces with a reduction to the canonical system of an associated cuspidal Gorenstein curve C', using theorems of Catanese [6]. The paper also sketches an explicit cusp-resolution step (deferred to the companion paper [39]) and reports on a SageMath implementation with random experiments for p=2,3,5.","tokens_in":25010,"tokens_out":8251,"duration_ms":88505,"significance":"If Theorem 3.1 is correct, it provides a clean geometric explanation of the hyperelliptic/non-hyperelliptic dichotomy for genus-3 stable reduction and gives a practical route to explicit stable models in the non-hyperelliptic case, including the difficult case p=2. The residue-theoretic framework is promising, and the companion implementation with archived random experiments is a valuable stress test. The paper is, however, not self-contained: the proof of the main theorem relies heavily on the master's thesis [29] and on Catanese's theorems [6], especially in characteristic 2, where the cited results involve subtleties that the manuscript does not verify. These gaps are load-bearing for the central claim and require attention before the theorem can be considered established.","major_comments":[{"comment":"The proof of the forward direction of Theorem 3.1 hinges on the assertions that C' is '2-inseparable' and 'very strongly connected' and that |ω_{C'/k}| is base-point-free by [6, Theorem D]. However, Definition 1.5 defines 2-inseparability only for semistable curves with ordinary double points, whereas C' has A2 cusps. The statement 'C' is constructed from the 2-inseparable core, so it is very strongly connected' is an assertion, not a proof. Since [6, Theorems D, F, G] require very strongly connected (and non-hyperelliptic) hypotheses, this transfer is load-bearing. Please either prove that the Gorenstein curve obtained in Lemma 3.7 from a 2-inseparable core is very strongly connected in the sense of [6, Definition 3.21], or give a precise reference that establishes this implication.","section":"§3.2, paragraphs after Lemma 3.8"},{"comment":"The characteristic-2 case is the main motivation of the paper, but the argument does not verify that Catanese's theorems apply in characteristic 2. Lemma 3.9 establishes only the non-existence of a finite degree-2 morphism C' → P^1, i.e. that C' is not 'honestly hyperelliptic' in the terminology of [6, Definition 3.18]. The conclusion that C' is not 'hyperelliptic' in the sense needed for [6, Theorem G] uses [6, Theorem F] together with the unproved assertion that C' is very strongly connected. Since [6] itself distinguishes the two notions, and the present paper's Section 1.2 documents special behavior in characteristic 2 (e.g. Proposition 1.9), the equivalence is not automatic. The authors should provide a concrete verification, or a reference, that [6, Theorem F] holds for A1/A2-Gorenstein curves over algebraically closed fields of characteristic 2. Without this, the canonical map may","section":"§3.2, final paragraph and Lemma 3.9"},{"comment":"The proof of Proposition 3.6 reduces the crucial dimension statement dim(W_Res)=3 to 'an elementary Riemann-Roch calculation' deferred to [29, Lemma 3.44]. This independence of the residue conditions is load-bearing: if the constraints were not independent, the constructed linear series would have dimension less than 3 and would not define a plane model. The main theorem should not rest on an unstated computation from a separate thesis. Please include the calculation in the paper or reproduce [29, Lemma 3.44] with proof.","section":"§3.2, Proposition 3.6"},{"comment":"The effective/computational claim of the paper rests on Theorem 3.11, quoted from the forthcoming companion paper [39], and Lemma 3.10, whose proof cites [22]. This is acceptable in a series, but the present paper should state explicitly how much of the proof of Theorem 3.1 is self-contained. In particular, the uniqueness and properties of the minimal semistable model dominating X0 in Lemma 3.10 are quoted rather than proved. If the converse direction of Theorem 3.1 is intended as a new contribution, the dependence on [22] and on [39] should be spelled out precisely; otherwise the reader cannot separate the new arguments from imported results.","section":"§3.4, Theorem 3.11 and §3.3, Lemma 3.10"}],"minor_comments":[{"comment":"The phrase 'By (1), H^0(C', ω_{C'/k}) is precisely the subspace...' is ambiguous: equation (1) in the paper is the cusp normal form from Lemma 2.2, not the residue condition from Proposition 3.6. Please renumber or rephrase.","section":"§3.2, Lemma 3.8"},{"comment":"The classification table uses line-thickness conventions for geometric genus that are not visible in the text version. Please include an explicit graph-label key or refer to [35, Figure 2.2] for each entry.","section":"§1.3, Proposition 1.13"},{"comment":"The term '2-inseparable' is defined only for semistable curves, but later applied to the Gorenstein curve C' with A1/A2 singularities. Please state the definition for this broader class or clearly indicate the intended meaning when transferring to C'.","section":"Definition 1.5 and §3.2"},{"comment":"References [32] and [39] are listed as 'forthcoming'/'2026'. Since Theorem 3.11 and parts of the implementation depend on them, a stable arXiv identifier or a preprint link would help the reader verify the claims.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and the proposed dichotomy is attractive. My main concern is that the proof of the central theorem depends on several unverified transfers to Catanese's framework, particularly in characteristic 2, and on key statements deferred to the master's thesis [29]. The authors should be asked to either prove these steps or make the dependence explicit with precise references. This is not a rejection: the strategy is plausible and the implementation suggests the theorem is true. However, the current version does not give the reader enough evidence to certify the main claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a worthwhile paper with a clean main theorem and working code, but the proof of the forward direction has a genuine gap that needs to be closed before the theorem is fully trusted. The gap is exactly where the stress-test says it is: the appeal to Catanese's base-point-free and closed-immersion theorems for the cusp curve C' in characteristic 2.\n\nWhat's genuinely new: the paper addresses a concrete hole in the explicit semistable-reduction toolkit. For smooth plane quartics in residue characteristic 2, the standard admissible-reduction routes fail, and this paper offers a GIT-based route that is actually implemented. The dichotomy (non-hyperelliptic stable reduction iff unique GIT-stable plane model) is a clean statement. The construction of the GIT-stable model via residues on the core (Proposition 3.6, Lemma 3.8) is a good piece of local geometry, and the converse direction from a GIT-stable model to a stable model is standard but clearly written. The implementation and the random experiments are reproducible, with data and code archived; that is real evidence. The reported failure/timeout rates are also honest.\n\nThe soft spots: First, the paper freely states that Sections 1 and 3 are based on the first author's master thesis [29], and the main proof quotes key lemmas from it. On its own this is not a flaw -- the thesis is cited -- but it means the present paper is not self-contained. Second, and more substantively, the forward direction of Theorem 3.1 uses Catanese's theorems [6, D, F, G] to conclude that |ω_{C'}| is base-point-free and gives a closed immersion. The paper asserts that C' is '2-inseparable' and hence 'very strongly connected' without showing that the 2-inseparability definition, which is made for nodal semistable curves, transfers to a curve with A2 cusps. And it does not check that Catanese's results hold in characteristic 2 for Gorenstein curves with these singularities. Lemma 3.9 only rules out a degree-2 map to P^1; the step from that to 'not hyperelliptic in Catanese's sense' relies on Theorem F, and if that equivalence fails the whole construction could collapse. This is the part I'd want a referee to verify line by line.\n\nThe paper is not circular and there is no fake data. The question is whether the gap is a quick fix (likely: a longer argument in [29] or a characteristic-2 check) or a real obstruction. I don't know which, and that's the point.\n\nWho this is for: anyone working on explicit arithmetic of genus-3 curves over local fields, especially in wild residue characteristic. It deserves a serious referee -- an editor should send it out with a request to (a) state precisely what in [29] proves the missing steps, or include them, and (b) verify Catanese's hypotheses in char 2. My own verdict would be conditional, not accept-as-is.","headline":"A useful, concrete method for p=2 quartic reduction, but Theorem 3.1's forward direction rests on an unverified use of Catanese's theorems in characteristic 2.","tokens_in":25536,"tokens_out":3333,"would_cite":false,"duration_ms":33884,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G20","11G20","14H25","14H50","14L24","14Q25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For non-hyperelliptic genus-3 curves, stable reduction is exactly a unique GIT-stable plane model with each cusp replaced by a genus-one tail.","keywords":["semistable reduction","plane quartics","GIT-stable models","stable curves genus 3","1-tails","cusps","residue characteristic 2","weighted blow-up"],"falsifier":"Compute, over a 2-adic field, a smooth plane quartic whose stable special fibre is a 2-inseparable non-hyperelliptic core with one tail, and run the paper's algorithm: the special fibre of the resulting plane model should be reduced with only nodes and cusps. If it contains a tacnode or a triple point, the canonical embedding of C' is not a closed immersion, disproving the theorem. Alternatively, exhibit a characteristic-2 genus-3 stable curve whose associated cusp curve C' has a canonical system with a base point.","tokens_in":24433,"feed_emoji":"📐","tokens_out":4569,"duration_ms":48818,"temperature":0.7,"pith_summary":"This paper establishes a precise criterion for when semistable reduction of a smooth plane quartic can be read off from a GIT-stable plane model: such a model exists and is unique exactly when the stable reduction is not hyperelliptic. In that case the stable model is obtained from the plane model by replacing each cusp of the special fibre with a genus-one tail, and the contraction map from the stable model to the plane model contracts precisely those tails. The proof isolates the mechanism behind the hyperelliptic/non-hyperelliptic dichotomy through canonical linear series on a Gorenstein curve built from the core of the stable curve. The criterion covers all residue characteristics, including the previously inaccessible case p=2, and it is made algorithmic.","feed_headline":"Non-hyperelliptic quartic reduction lives on a unique plane model","feed_subtitle":"The stable model is the plane model with each cusp replaced by a genus-one tail, and now it is computable at p = 2.","key_machinery":"The central object is the pair (stable model, GIT-stable plane model) connected by a contraction. The proof works with the dualizing sheaf twisted by the tail divisors, D = Σ d_i X_i, showing that the resulting linear series contracts the tails and restricts on the core to the canonical system of a Gorenstein curve C' with at most A1/A2 singularities. Residue reciprocity on the arithmetic surface identifies the relevant sections as those with vanishing residues at the attachment points, so the canonical map of C' becomes the special fibre of the plane model. The constructive side uses a weighted blow-up of the ideal (Π,x,y) with weights (1,2,3) at each cusp, whose exceptional divisor is an e","core_discovery":"Theorem 3.1: if a smooth non-hyperelliptic genus-3 curve over a discretely valued field has semistable reduction, its stable model dominates a unique GIT-stable plane model; the dominant map is an isomorphism away from the cusps of the special fibre and contracts each 1-tail of the stable fibre to a cusp. Conversely, if a GIT-stable plane model exists, the stable reduction is non-hyperelliptic, and the stable model is the minimal semistable model dominating the plane model, obtained by resolving each cusp into a genus-one tail. The dichotomy is detected by a Gorenstein curve C' formed from the stable fibre by smashing each tail attachment point into a cusp: the plane model exists exactly whe","pith_inferences":["The same picture—form a cuspidal Gorenstein curve from the core, then resolve cusps to tails—suggests an approach to semistable reduction of non-hyperelliptic curves of higher genus via GIT-stable projective models; the bottleneck would be the analogous base-point-freeness and embedding statement in every characteristic.","The Ciani example shows that the extension needed for a GIT-stable model can be larger than the minimal extension for semistable reduction; a more economical algorithm might choose the GIT extension more carefully.","The characteristic-2 classification predicts that a genus-3 stable fibre with three 1-tails attached to a genus-zero core is automatically hyperelliptic in characteristic 2, so such quartics will be exactly the strictly semistable cases inaccessible to this method.","A concrete testable consequence: if the imported characteristic-2 theorems on canonical systems fail, the first counterexample should appear as a plane quartic whose computed special fibre contains a tacnode or worse singularity, which the implementation could flag directly."],"forward_implications":["For non-hyperelliptic stable reduction, the stable model is completely described: the GIT-stable plane model plus, for each cusp, one genus-one tail attached along a node.","The criterion gives a necessary and sufficient dichotomy: hyperelliptic stable reduction is exactly the case in which no GIT-stable plane model exists and the GIT model is only strictly semistable.","Once a GIT-stable plane model is known over some extension, a further finite extension admits the stable model as a modification of that plane model, so explicit cusp resolution computes the stable model.","The method covers residue characteristic 2, the case not handled by cover-based algorithms; in characteristic 2 the hyperelliptic case acquires extra components on which the involution acts trivially, corresponding to relative Frobenius.","Random experiments for p = 2, 3, 5 produce explicit reduction types for hundreds of quartics, showing that the two-step pipeline terminates and is practical on large samples."],"fun_headline_variants":["Computing stable reduction of quartics now works at p=2","Stable reduction of quartics: plane model replaces cusps with tails","Non-hyperelliptic quartics: stable model from plane model via cusps","Quartic stable reduction reduces to plane model cusp resolution"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof leans on a classification of mildly singular Gorenstein curves in characteristic 2 that is taken from earlier literature; if the relevant distinction between two notions of hyperellipticity collapses in characteristic 2, the plane model constructed under the non-hyperelliptic assumption could turn out to be only semistable rather than stable.","fun_headline_variants_meta":{"raw":{"variants":["Computing stable reduction of quartics now works at p=2","Stable reduction of quartics: plane model replaces cusps with tails","Non-hyperelliptic quartics: stable model from plane model via cusps","Quartic stable reduction reduces to plane model cusp resolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1164,"prompt_tokens":724,"completion_tokens":440,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":361}},"tokens_in":468,"tokens_out":440,"duration_ms":4747,"temperature":1.0,"reasoning_tokens":361,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T11:34:47.602061+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, over a 2-adic field, a smooth plane quartic whose stable special fibre is a 2-inseparable non-hyperelliptic core with one tail, and run the paper's algorithm: the special fibre of the resulting plane model should be reduced with only nodes and cusps. If it contains a tacnode or a triple point, the canonical embedding of C' is not a closed immersion, disproving the theorem. Alternatively, exhibit a characteristic-2 genus-3 stable curve whose associated cusp curve C' has a canonical system with a base point.","supporting_citations":[],"review_version":1}