{"id":"73923759-1bb9-420b-8ada-7593892a556e","arxiv_id":"2507.17633","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Every Q-Gorenstein smoothable normal stable Horikawa surface falls into one of seven explicit families, and its global smoothability is controlled by a single local condition at its elliptic double cone singularities.","lead":"This paper classifies all normal stable Horikawa surfaces, a family of complex surfaces of general type, whose singularities are locally smoothable, and gives a criterion for global smoothability of each. It also describes the boundary strata of their moduli space, with one explicitly excluded pg=10 case, and determines exactly when that moduli space is connected.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.7's construction of a good involution via the relative canonical double cover fails for non-standard Horikawa surfaces, whose canonical map is composed with a pencil; the classification of all Q-Gorenstein smoothable surfaces therefore rests on an unproven reduction.","rationale":"The reader identified the log Noether inequality (Theorem 4.2) as the weakest assumption. That is a reasonable structural concern, but the paper provides a proof of Theorem 4.2 and the trichotomy of Proposition 5.3 is carefully derived. The step I find more load-bearing is Proposition 5.7, because it is the bridge from 'Q-Gorenstein smoothable' to 'admits a good involution', and its proof contains a concrete false assertion for non-standard Horikawa surfaces: the relative canonical morphism need not be a double cover. If this bridge fails, Theorem 1.3 is not a classification of all smoothable surfaces but only of smoothable surfaces with a good involution, and Theorem 1.5's equivalence is likewise restricted. The concern is not that the theorem is false; the hyperelliptic involution of the canonical pencil may well supply the missing argument, and the paper's explicit constructions do produce good involutions. But the proof as written does not supply that argument, so the conditional verdict remains appropriate: the classification is credible but not fully established. I therefore recommend keeping the reader's CONDITIONAL verdict (UNCHANGED), while flagging that the specific gap is in Proposition 5.7 rather than the log Noether inequality.","tokens_in":62392,"tokens_out":13211,"duration_ms":135377,"concrete_test":"Let X→C be the Q-Gorenstein smoothing of a Lee-Park type Horikawa surface with pg=4 constructed in Section 9/Construction 7.53, and compute the relative canonical morphism X_{C\\{0}} → P_C(f_*ω_{X/C}) on the punctured family. If the general fiber X_c is non-standard, the image has dimension 2 and the map is a genus-2 fibration, confirming that Proposition 5.7's asserted double cover does not exist. To settle the mathematical claim, independently construct the hyperelliptic involution of the canonical pencil on the general fiber, verify it is independent of c, and use the valuative criterion for Aut_C(X) to extend it to the central fiber; if no extension exists, Theorem 1.3 is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reduction of Theorem 1.3 to surfaces with a good involution depends on Proposition 5.7, which asserts that every Q-Gorenstein smoothable stable Horikawa surface carries a unique good involution. In the proof, the authors take a Q-Gorenstein smoothing X→C with smooth general fibers and define σ^∘ as the covering involution of the relative canonical morphism φ_{X^∘/C}, stating: 'Let W^∘ denote the image of X^∘ under φ_{X^∘/C}. Then φ_{X^∘/C} : X^∘ → W^∘ is a double covering.' This is valid only for standard Horikawa surfaces. By Definition 5.1, a non-standard Horikawa surface has its canonical linear system composed with a pencil, so the canonical map of a smooth general fiber X_c has one-dimensional image; the relative canonical morphism from the threefold to the relative projective space then has image of dimension two and one-dimensional fibers, i.e., a genus-2 fibration, not a finite double cover. The covering involution σ^∘ is therefore not defined by this argument. The later portion of the proof treats the non-standard case separately, but it assumes the existence of the involution (e.g., 'the involution eσ induced by σ') rather than constructing it from the canonical map. Since Section 1.4 similarly asserts that every Q-Gorenstein smoothable surface has a good involution 'as the limit of the good involutions of smooth Horikawa surfaces', and for non-standard smooth Horikawa surfaces the relevant involution is the hyperelliptic involution of the canonical pencil, not the covering transformation of the canonical map, the proof as written leaves a gap. Without a repaired argument, the classification in Theorem 1.3 covers only those surfaces admitting a good involution, and the claim that it classifies all Q-Gorenstein smoothable normal stable Horikawa surfaces is not established.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies normal stable Horikawa surfaces (surfaces of general type with K^2 = 2p_g - 4) whose singularities are log canonical and Q-Gorenstein smoothable. The main theorem (Theorem 1.3) asserts that every such Q-Gorenstein smoothable surface falls into one of seven families: standard double covers of minimal degree surfaces; pg=4 Gorenstein surfaces with bicanonical double cover of an elliptic cone; Lee-Park type surfaces with two T-singularities; and four pg=3 configurations involving the T-singularity 1/50(1,29), the strictly lc singularity (2,2,2,2), and two 1/4(1,1) T-singularities. Theorem 1.4 reduces global Q-Gorenstein smoothability to equivariant smoothability of elliptic double cone singularities with respect to the good involution. The paper also describes the KSBA boundary strata (Theorem 1.14) and proves connectedness of the moduli space for pg=6, pg=10, and pg-2 not divisible by 4 (Theorem 1.17). The technical machinery includes a log Noether inequality, extended T-chains, good involutions, and anti-P-resolutions. The paper is very long (212 pages), with the classification and deformation proofs concentrated in Section 7, Section 9, and Appendices A-C.","tokens_in":62660,"tokens_out":5891,"duration_ms":67466,"significance":"If the classification is correct, it is a substantial advance: explicit descriptions of KSBA boundary strata for surface moduli are rare, and the paper gives a complete normal-locus boundary description for Horikawa surfaces except for one explicitly flagged supersingular pg=10 case. The reduction in Theorem 1.4 to local equivariant smoothability is clean and useful, and the anti-P-resolution machinery appears to be a new and potentially transferable tool. The paper is also honest about its limitations: Remark 1.2(2) states that existence is not verified for most candidates in the broader non-standard classification, and the pg=10 supersingular case is explicitly left partially open. The extensive case analysis is not mechanically verified, which sets a ceiling on confidence, but the parts I could check are careful in their attributions and caveats. The main weakness is a load-bearing gap in the proof of Proposition 5.7, on which the reduction to good involutions depends.","major_comments":[{"comment":"The proof of Proposition 5.7 states: 'Let W° denote the image of X° under φ_{X°/C}. Then φ_{X°/C}: X° → W° is a double covering.' This is valid only for standard Horikawa surfaces. For a non-standard Horikawa surface, Definition 5.1 says the canonical linear system is composed with a pencil, so the relative canonical morphism over C\\{0} has positive-dimensional generic fibers and one-dimensional image; it is not a finite double cover and does not define a covering involution σ°. The later portion of the proof treats the non-standard case, but it appears to assume the existence of the involution (for example, in forming W := X/σX and the fiberwise involution eσ) rather than constructing it from the smoothing family. Since Proposition 5.7 is the basis for the assertions in Section 1.4 and Theorem 1.3 that every Q-Gorenstein smoothable normal stable Horikawa surface admits a unique good involution, the reduction of the classification to surfaces with a good involution is incomplete as written.","section":"Section 5.2, Proposition 5.7"},{"comment":"Section 1.4 asserts that every Q-Gorenstein smoothable normal stable Horikawa surface carries the good involution 'as the limit of the good involutions of smooth Horikawa surfaces.' For smooth non-standard Horikawa surfaces the relevant involution is the hyperelliptic involution of the canonical pencil, not the covering involution of the canonical map; the text does not prove that this involution extends to the limit in the singular family. This is not merely a presentation issue: if the extended involution fails to exist or is not the one induced by the smoothing, then the list in Theorem 1.3, which is obtained by interspersing the good-involution classification with the smoothability analysis, could miss a family. The authors should either supply a direct construction of σ for the non-standard case or prove separately that every Q-Gorenstein smoothing of a non-standard Horikawa surface is equivariant with respect to the hyperelliptic involution of the general fibers.","section":"Section 1.4 and Theorem 1.3"}],"minor_comments":[{"comment":"There are several typographical errors: 'Quenstion' in Section 1.6, 'toransform' in the proof of Proposition 6.2, 'suffces' in Lemma 2.7, and the running title contains 'NOETHER-HORIKA W A' with a broken spacing. These should be corrected in the final version.","section":"Throughout"},{"comment":"The operation 'a, 2^{-1}, b' in a string is defined as a+b-2, but the example '[3,2^{-1},3] denotes [4]' is correct; however, the discussion would be clearer if the operation were described as a legal substring replacement rather than as a separate number inserted into the chain.","section":"Notation 3.1(2)"},{"comment":"Several figures, especially Figures 2, 6, and 7, are difficult to read in the arXiv version because of small labels and dense decorations. Since the stratification results in Section 9 refer to these figures repeatedly, higher-resolution or vector versions would materially help the reader.","section":"Figures 1-7"},{"comment":"The statement 'Except for a few cases, we do not verify the existence of a non-standard Horikawa surface whose set of non-Du Val singularities coincides with a given candidate' could be misunderstood as applying to Theorem 1.3. It applies to the broader classification Theorem 7.49; the text would be clearer if this distinction were repeated at the point of Theorem 7.49.","section":"Remark 1.2(2)"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the gap in Proposition 5.7, and I would like the editor to ensure that the authors address it explicitly. The paper is too long for a single referee to verify all case analysis; if the central reduction is fixed, a companion file with machine-checked or heavily detailed versions of the combinatorial classifications in Appendices A and C would substantially raise confidence. I do not see evidence of unacknowledged prior work; the attribution to Horikawa, Chen, Kollár-Shepherd-Barron, and the smoothability literature appears careful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, if the main theorem holds up, it completes the classification of Q-Gorenstein smoothable normal stable Horikawa surfaces and describes the KSBA boundary, which is real progress on a problem at least three groups are actively working on. Second, the proof of the key reduction, Proposition 5.7, has a gap for non-standard surfaces, and the classification as stated currently rests on it.\n\nWhat is genuinely new: the good-involution framework, the anti-P-resolution tool, the classification of the strictly lc cases, the first (2,4,4)[3] example, and the pg=4 family with bicanonical double cover of an elliptic cone. The authors are careful with attribution: Theorem 6.8 is Chen's, the pg=3 overlap with Evans-Simonetti-Urzúa is acknowledged, and the incomplete spots (existence of candidates in Theorem 7.49, pg=10 supersingular stratification) are flagged as such. The local-to-global smoothability criterion, Theorem 1.4, is a solid idea, and the combinatorial machinery of extended T-chains and P-admissibility looks reusable.\n\nThe soft spots. The main one is Proposition 5.7. The proof defines the involution as the covering involution of the relative canonical morphism. That works for standard fibers. For non-standard fibers, the canonical map is a pencil of genus-2 curves, so the relative canonical morphism has one-dimensional fibers, not a finite double cover, and no covering involution is produced. The later non-standard part of the proof presumes the involution already exists. Section 1.4 asserts existence as the limit of good involutions of smooth Horikawa surfaces, which is plausible since smooth non-standard Horikawa surfaces have the hyperelliptic involution of the canonical pencil, but that argument is not written. So Theorem 1.3, as proven, covers only surfaces admitting a good involution. That is load-bearing, though likely repairable.\n\nThe other caveats are less severe. The long case analyses in Sections 7.2-7.7 and 9 are unformalized; I trust the parts I checked, but nobody can certify 150 pages of hand computation. Existence of the Theorem 7.49 candidates is not established, only recipes. The pg=10 supersingular case is left open, honestly.\n\nWho is this for? Specialists in surface geography and KSBA moduli. It deserves a serious referee: the claims are important, the attribution is fair, and the tools are new. But the referee should require a corrected proof of Proposition 5.7 before the headline classification is accepted; without that, Theorem 1.3 is conditional.","headline":"A large, mostly credible classification of stable Horikawa degenerations, but the proof of the key good-involution reduction has a real gap for non-standard surfaces, so the headline theorem is not yet fully established.","tokens_in":63539,"tokens_out":4817,"would_cite":false,"duration_ms":51211,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29","14J17","14B07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies all normal stable Horikawa surfaces with only $\\mathbb{Q}$-Gorenstein smoothable log canonical singularities, proves a local-to-global criterion for their $\\mathbb{Q}$-Gorenstein smoothability, and describes the…","keywords":["Horikawa surfaces","stable surfaces","log canonical singularities","Q-Gorenstein smoothability","moduli space","KSBA compactification","T-singularities","elliptic fibrations"],"falsifier":"Compute the invariant triple $(g,n_X,l_{Eh})$ for the minimal resolution of any purported counterexample: Proposition 5.3(2) says a Q-Gorenstein smoothable normal stable Horikawa surface must fall into exactly three cases, so a surface with an elliptic fibration over $\\mathbb{P}^1$ with one horizontal component and one elliptic singularity would immediately refute the classification. A second check is the singularity list: any Q-Gorenstein smoothable normal stable Horikawa surface whose non-Gorenstein singularities are not among the seven configurations in Theorem 1.3 would also refute it.","tokens_in":61999,"feed_emoji":"","tokens_out":9673,"duration_ms":95715,"temperature":0.7,"pith_summary":"The paper aims to complete the classification of normal stable Horikawa surfaces, the surfaces of general type on the Noether line $K_X^2=2p_g(X)-4$. It asserts that among such surfaces with only $\\mathbb{Q}$-Gorenstein smoothable log canonical singularities, exactly seven families occur: Gorenstein double covers of minimal degree surfaces, $p_g=4$ surfaces whose bicanonical map double-covers an elliptic cone, Lee-Park type surfaces with two T-singularities of type $\\frac{1}{(p_g-1)^2}(1,p_g-2)$, and four $p_g=3$ configurations built from $\\frac{1}{50}(1,29)$, the strictly lc singularity $(2,2,2,2)$, and two $\\frac{1}{4}(1,1)$ T-singularities. It then reduces global $\\mathbb{Q}$-Gorenstein smoothability to a local condition: every elliptic double cone singularity must be equivariantly smoothable with respect to the unique good involution. If correct, the boundary of the KSBA moduli space of normal stable Horikawa surfaces is explicitly stratified, and the moduli space is connected exactly when $p_g=6$, $p_g=10$, or $p_g-2$ is not divisible by $4$.","feed_headline":"Seven singularity families cover all smoothable Horikawa degenerations","feed_subtitle":"A local criterion on their singularities decides which stable Horikawa surfaces lie on the moduli boundary.","key_machinery":"The argument is carried by four objects. The log Noether inequality (Theorem 4.2) is a logarithmic version of Noether's inequality for normal stable surfaces: it replaces the surface by its minimal resolution together with the reduced exceptional divisor over its elliptic singularities, and it forces every non-standard Horikawa surface's minimal model to be an elliptic fibration over a rational or elliptic curve, organized by the trichotomy $(g,n_X,l_{Eh})=(0,1,0),(0,2,0),(1,1,1)$. Extended T-chains are combinatorial records of exceptional curves that arise by contracting T-chains separated by $(-1)$-curves; their P-admissibility test decides which singularity configurations can actually occur. A good involution is the unique involution that a smoothable normal stable Horikawa surface inherits as the limit of the covering involution of the canonical map on smooth fibers, and it lets the paper replace the surface by the quotient pair $(W,\\frac{1}{2}B)$. An anti-P-resolution is a birational modification $W^-\\to W$ with $\\mathbb{Q}$-Gorenstein smoothable slc singularities and ample relative anticanonical divisor; it is the device that makes the relevant cohomology vanish and turns local equivariant smoothability into a global smoothing.","core_discovery":"The central claim is Theorem 1.3: every $\\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surface belongs to exactly one of seven families. They are the Gorenstein standard surfaces whose canonical map is a double cover of a minimal-degree surface; the $p_g=4$ Gorenstein surfaces whose bicanonical map is a double cover of an elliptic cone; Lee-Park type surfaces with two T-singularities of type $\\frac{1}{(p_g-1)^2}(1,p_g-2)$; and four $p_g=3$ configurations assembled from the T-singularity $\\frac{1}{50}(1,29)$, the strictly lc singularity $(2,2,2,2)$, and two $\\frac{1}{4}(1,1)$ T-singularities. The companion Theorem 1.4 says a surface in this list is globally $\\mathbb{Q}$-Gorenstein smoothable exactly when all its elliptic double cone singularities are equivariantly smoothable with respect to the good involution; in particular, surfaces with no elliptic double cone singularity are automatically smoothable. From this the paper derives an explicit stratification of the boundary of the moduli space of $\\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surfaces, and proves that the moduli space is connected precisely when $p_g=6$, $p_g=10$, or $p_g-2$ is not divisible by $4$.","pith_inferences":["The same log Noether inequality and extended T-chain analysis should classify normal stable surfaces on the next line above the Horikawa line, namely surfaces with $K^2=2p_g-3$; the paper sketches this as its planned continuation.","Anti-P-resolutions give a general technique for equivariant cusp smoothing when a finite group acts non-freely outside the singularity, a setting broader than the free-action cases treated in earlier deformation theory.","The $p_g=10$ connectedness result reopens the Horikawa problem: the two Gieseker components can now be joined through lc degenerations, so the remaining question is whether such lc degenerations preserve diffeomorphism type in the same way that degenerations through T-singularities do.","Because the good involution is unique on every smoothable normal stable Horikawa surface, the quotient-pair viewpoint is intrinsic rather than a choice: any smoothable surface must carry this symmetry, so no surface is missed by studying double covers of quotients."],"forward_implications":["Because Theorem 1.4 reduces global smoothability to a local condition, a Horikawa surface with no elliptic double cone singularity is automatically Q-Gorenstein smoothable, and one with such a singularity is smoothable exactly when that singularity is equivariantly smoothable.","The seven-family classification means that the boundary of the moduli space of Q-Gorenstein smoothable normal stable Horikawa surfaces is stratified in the explicit way the paper diagrams, with the degeneration behavior of Lee-Park and standard type surfaces depending on the geometric genus.","The moduli space is connected exactly for $p_g=6$, $p_g=10$, or $p_g-2$ not divisible by $4$; in particular, the two Gieseker components for $p_g=10$ are joined through supersingular standard Horikawa surfaces.","The list includes the first known stable Horikawa surface with a strictly lc singularity of type $(2,4,4)[3]$, constructed through the paper's recipe from an elliptic fibration with prescribed singular fibers.","For $p_g\\geq 4$, among non-Gorenstein klt Horikawa surfaces only those of Lee-Park type can be Q-Gorenstein smoothable, sharpening earlier classifications of the klt case."],"supporting_citations":[{"why":"Supplies the logarithmic Noether inequality and the classification of standard Horikawa surfaces as double covers of minimal degree surfaces, which Theorem 6.8 extends.","marker":"[32]"},{"why":"Provides the classification of T-chains, P-resolutions, and Q-Gorenstein smoothable slt singularities that underpin the extended T-chain analysis and the smoothing criteria.","marker":"[97]"},{"why":"Horikawa's original classification of surfaces on the Noether line and the two-component phenomenon for $p_g=10$ are the baseline that Theorems 1.14 and 1.17 extend to lc normal stable surfaces.","marker":"[76]"},{"why":"Shows that stable Horikawa surfaces arising as double covers of Hirzebruch surfaces need not be Q-Gorenstein smoothable, motivating the global smoothability question answered by Theorem 1.4.","marker":"[127]"},{"why":"Classified klt non-Gorenstein stable Horikawa surfaces with only T-singularities, the case that Theorem 1.3 and Theorem 7.49 extend to log canonical singularities.","marker":"[114]"},{"why":"Introduced the Lee-Park klt Horikawa surfaces with two Wahl singularities that are the prototypes for the Lee-Park type family in Theorem 1.3(3).","marker":"[104]"},{"why":"Established the $p_g=3$ case of Q-Gorenstein smoothable normal stable Horikawa surfaces, which is subsumed and generalized by Theorems 1.3 and 1.5.","marker":"[51]"},{"why":"Fujino's vanishing theorem for slc schemes is used to prove the cohomology vanishing that turns local equivariant smoothability into global Q-Gorenstein smoothability.","marker":"[60]"},{"why":"The second author's Z-positive vanishing theorem is used when Fujino's vanishing theorem is inapplicable, notably for supersingular standard Horikawa surfaces with $p_g\\neq 10$.","marker":"[48]"}],"fun_headline_variants":["Seven families classify all Q-Gorenstein smoothable Horikawa surfaces","Local criterion decides global smoothability of Horikawa surfaces","Boundary of Horikawa moduli space fully stratified","Seven singularity types describe all normal stable Horikawa deformations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the log Noether inequality for normal stable surfaces: if some log canonical surface with elliptic singularities escapes the trichotomy it imposes, the seven-family list would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Seven families classify all Q-Gorenstein smoothable Horikawa surfaces","Local criterion decides global smoothability of Horikawa surfaces","Boundary of Horikawa moduli space fully stratified","Seven singularity types describe all normal stable Horikawa deformations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2561,"prompt_tokens":885,"completion_tokens":1676,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":1609}},"tokens_in":501,"tokens_out":1676,"duration_ms":14645,"temperature":1.0,"reasoning_tokens":1609,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:48:04.199129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the invariant triple $(g,n_X,l_{Eh})$ for the minimal resolution of any purported counterexample: Proposition 5.3(2) says a Q-Gorenstein smoothable normal stable Horikawa surface must fall into exactly three cases, so a surface with an elliptic fibration over $\\mathbb{P}^1$ with one horizontal component and one elliptic singularity would immediately refute the classification. A second check is the singularity list: any Q-Gorenstein smoothable normal stable Horikawa surface whose non-Gorenstein singularities are not among the seven configurations in Theorem 1.3 would also refute it.","supporting_citations":[{"cited_title":"The minimal and next minimal volumes of normal KSBA stable surfaces with $p_g\\ge 2$","cited_arxiv_id":"2308.01473","evidence_quote":"Supplies the logarithmic Noether inequality and the classification of standard Horikawa surfaces as double covers of minimal degree surfaces, which Theorem 6.8 extends."},{"cited_title":"Koll´ ar,Families of varieties of general type, Cambridge Tracts in Mathematics 231","cited_arxiv_id":null,"evidence_quote":"Provides the classification of T-chains, P-resolutions, and Q-Gorenstein smoothable slt singularities that underpin the extended T-chain analysis and the smoothing criteria."},{"cited_title":"Pearlstein, Z","cited_arxiv_id":null,"evidence_quote":"Shows that stable Horikawa surfaces arising as double covers of Hirzebruch surfaces need not be Q-Gorenstein smoothable, motivating the global smoothability question answered by Theorem 1.4."},{"cited_title":"Classification of Horikawa surfaces with T-singularities","cited_arxiv_id":"2410.02943","evidence_quote":"Classified klt non-Gorenstein stable Horikawa surfaces with only T-singularities, the case that Theorem 1.3 and Theorem 7.49 extend to log canonical singularities."},{"cited_title":"Konno, Chain-connected component decomposition of curves on surfaces, J","cited_arxiv_id":null,"evidence_quote":"Introduced the Lee-Park klt Horikawa surfaces with two Wahl singularities that are the prototypes for the Lee-Park type family in Theorem 1.3(3)."},{"cited_title":"Evans, A","cited_arxiv_id":null,"evidence_quote":"Established the $p_g=3$ case of Q-Gorenstein smoothable normal stable Horikawa surfaces, which is subsumed and generalized by Theorems 1.3 and 1.5."}],"review_version":1}