{"id":"f96e883f-8898-4327-a843-fbb89e523ada","arxiv_id":"1908.07928","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves the Morrison-Kawamata cone conjecture and finite generation of automorphisms for Enriques surfaces in odd characteristic, and constructs a positive-characteristic surface birational to an Enriques surface with discrete non-finitely generated automorphism group.","lead":"This paper proves that every Enriques surface in odd characteristic has a finitely generated automorphism group and that this group acts on the divisor cone with a simple polyhedral fundamental domain. It also constructs new smooth surfaces in odd characteristic that are birational to Enriques surfaces but have automorphism groups that are discrete and not finitely generated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3 rests on Proposition 4.4, whose proof is only a sketch; the non-finite-generation conclusion depends on an unverified automorphism with f^2(x)=t^2x.","rationale":"The reader's finite-height/citation concern is legitimate but, in my view, secondary: over Enriques surfaces in characteristic 0 the automorphism group scheme is zero-dimensional, so a Lefschetz-principle argument would likely transfer Dolgachev's theorem from C to any algebraically closed field of characteristic 0; the paper should say this, but it is probably addressable. The more concrete and admitted gap is Proposition 4.4. It is the only place in Section 4 where non-finite-generation of the image of rho is established, and the proof is a sketch referring to an external paper. Without a full characteristic-p verification, Theorem 4.3 is unsupported as written. Since the gap is a missing proof rather than a demonstrated error, I keep the reader's CONDITIONAL verdict: the paper is conditionally acceptable pending a complete proof of Proposition 4.4 or an explicit replacement argument in odd characteristic.","tokens_in":985,"tokens_out":1014,"duration_ms":199679,"concrete_test":"Expand Proposition 4.4 into a complete proof in the paper itself. In particular verify: (i) the fibrations phi_{M_i}: X -> P^1 have no reducible fibers other than M_i over K in characteristic p; (ii) [Ma19, Lemma 2.17] supplies the stated automorphism g in characteristic p; (iii) the element f constructed by the [KO19] argument is defined over K and satisfies f^2(x) = t^2x on H2, with f^2 in Dec(X,Q32). If any cited lemma is only available in characteristic zero, or if the descent changes the action on H2, then Theorem 4.3 should be withdrawn or re-proved. A useful secondary check is to work out the construction for one explicit prime, such as p = 5, and verify that the image of rho contains a non-finitely generated subgroup.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper proves Theorem 4.3 by producing a representation rho: Ine(X,Q32,TQ32) -> (K,+) and then invoking Proposition 4.4 to assert that the image contains the additive subgroup M generated by {t^{-2n}a | n in Z_{>=0}}, which is not finitely generated. That assertion is the load-bearing step for the non-finite-generation of Ine(X,Q32,TQ32) and hence for Theorem 1.4. The proof of Proposition 4.4 is explicitly only a sketch: after naming two genus-one fibrations, it says that 'the same argument as in [KO19]' leads to an element f in Aut(X) with f^2(x) = t^2x on H2 and f^2 in Dec(X,Q32). No verification is provided that the cited results [Ma19, Lemma 2.17], [Ne64, Section III.17], and [Ta75, Section 6] have the same consequences in characteristic p > 3, nor that the descent from the characteristic-zero construction in [KO19] preserves the exact formula f^2(x) = t^2x. If any of these replacements fails, the image of rho need not contain M, and the non-finite-generation conclusion has no basis. The manuscript itself flags this by calling the proof a sketch, but the theorem nevertheless depends on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves the Morrison–Kawamata cone conjecture for Enriques surfaces over an algebraically closed field of odd characteristic, and shows that their automorphism groups are finitely generated. The proof passes to the K3 cover: for supersingular covers it uses Ogus's crystalline Torelli theorem and Lieblich–Maulik's Proposition 5.2, while for finite-height covers it uses Jang's lifting theorem and a characteristic-zero result of Dolgachev. In the second half, the paper constructs, for p>3, a blow-up of a specific Enriques surface whose automorphism group is discrete but not finitely generated, adapting the Keum–Oguiso complex-surface construction.","tokens_in":12809,"tokens_out":5986,"duration_ms":63901,"significance":"If the proofs are completed, Theorem 1.2 settles the cone conjecture for Enriques surfaces in odd characteristic and Theorem 1.3 gives the expected finite-generation result. Theorem 1.4 provides a positive-characteristic analogue of the Dinh–Oguiso and Keum–Oguiso examples and answers a question of [KO19]. The main outline is coherent and the reductions to published results are appropriate: the use of Ogus's crystalline Torelli theorem for supersingular K3 surfaces, the lifting arguments for finite-height K3 surfaces, and the arithmetic-group arguments for finite generation are all natural and, in the supersingular branch, essentially complete. The paper is clearly written and the strategy is convincing where full proofs are supplied.","major_comments":[{"comment":"Proposition 4.4 is the sole source of the claim that the image of the representation rho contains the subgroup generated by {t^{-2n}a | n>=0}, and its proof is explicitly only a sketch: it states that replacing [Ko86, Lemma 2.6] and [Ko63, Theorem 9.1] by [Ma19, Lemma 2.17], [Ne64, Section III.17], and [Ta75, Section 6], 'the same argument as in [KO19, Pages 10 and 11] leads to' an element f with f^2(x)=t^2x on H2. No verification is provided that these characteristic-p replacements yield the identical formula, nor that the descent from the characteristic-zero construction in [KO19] preserves the exact equality f^2(x)=t^2x. This equality is the load-bearing step for Theorem 4.3 and therefore for Theorem 1.4; the manuscript's own label 'Sketch of Proof' confirms that the essential verification is absent and must be supplied before the theorem can be accepted.","section":"Section 4, Proposition 4.4"},{"comment":"The finite-height branch of Theorem 3.6(3) invokes 'the main theorem of [Do84]' to conclude that W0⋊Aut(X0)* has finite index in O(L)+, with the only stated justification being that X0 is defined over an algebraically closed field of characteristic 0. If [Do84] is proved only for complex Enriques surfaces, a Lefschetz-principle argument or a more precise reference is needed to cover X0 over an arbitrary algebraically closed field of characteristic 0. This step is load-bearing for Theorems 1.2 and 1.3 in the non-supersingular case, so the exact scope of the cited result should be stated explicitly.","section":"Section 3, proof of Theorem 3.6(3)"}],"minor_comments":[{"comment":"The phrase 'rational polynomial fundamental domain' should read 'rational polyhedral fundamental domain'.","section":"Proof of Theorem 1.2"},{"comment":"The proposition is stated as a result but its proof is a sketch ending with 'The assertion then follows clearly'; either a complete proof should be given or the statement should be labeled as conditional on the cited characteristic-p analogues.","section":"Section 4, Proposition 4.4"},{"comment":"The text repeatedly refers to Figure 1 and to the curves Ei, Fi, and Cij, but no figure appears in the manuscript; if the figure is missing from the submitted version, it should be included, since the labels are essential for following Construction 4.2 and Theorem 4.3.","section":"Section 4, Figure 1"},{"comment":"Remark 1.5 says that for p=3 the desired example 'should' be obtainable by the same construction; please clarify whether Theorem 1.4 is intended to include p=3 or only p>3, and if p=3 is left open, state this explicitly in the main theorem.","section":"Remark 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's acknowledgments thank the referee for pointing out 'some gaps with very constructive remarks,' and the in-text label 'Sketch of Proof' on Proposition 4.4 is consistent with that. The main unresolved risk is whether the characteristic-p verification of the Keum–Oguiso construction actually goes through; an expert check of the cited [Ma19], [Ne64], and [Ta75] replacements is advisable. If the author can supply the missing details, the paper would be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The part worth knowing: the main theorem (cone conjecture plus finite generation of Aut for Enriques surfaces over algebraically closed fields of odd characteristic) is a clean reduction to known results, and I think it is basically right. Section 3 splits into supersingular K3-covers, handled by Ogus's crystalline Torelli theorem and Lieblich-Maulik's finite-index statement, and finite-height covers, handled by lifting to characteristic zero via Jang and then quoting Dolgachev. The argument follows Oguiso-Sakurai as the author says; there is no circularity and no invented input. The new content is real: Theorems 1.2 and 1.3 are not in the prior literature for odd characteristic, and the corollaries on finitely many rational curves and elliptic fibrations up to automorphism are natural payoffs. Citation patterns look honest; this is mainly assembly of known theorems, but assembly with a genuinely new conclusion.\n\nThe soft spot is Section 4. Proposition 4.4 is the load-bearing step for the non-finitely-generated example, and its proof is explicitly a sketch. The stress-test note lands: Theorem 4.3 needs an automorphism f with f^2(x)=t^2x on H2, and the paper does not verify that the characteristic-zero construction in [KO19] descends with the exact formula through the cited replacements [Ma19], [Ne64], [Ta75]. This is an addressable gap rather than a demonstrated error, but it is enough to keep Theorem 1.4 conditional. A referee should ask for a complete proof of Proposition 4.4, or at least a precise statement of which [KO19] lemmas survive in characteristic p>3.\n\nA smaller point: the finite-height branch of Theorem 3.6(3) leans on Dolgachev's finite-index theorem for the lifted characteristic-zero Enriques surface, and on Jang's lifting theorem preserving the Neron-Severi lattice. Both are reasonable, and the reader's worry here is minor; the reduction is coherent.\n\nWho this is for: anyone working on the Morrison-Kawamata cone conjecture in positive characteristic, or on automorphism groups of Enriques-like surfaces. The paper deserves a serious referee even with the Section 4 gap; the main theorems are important enough that I would not desk-reject it. My own verdict on Theorems 1.2 and 1.3 is fairly positive; on Theorem 1.4, I would want the gap closed before citing it as established.","headline":"A credible reduction proving the cone conjecture and finite generation for Enriques surfaces in odd characteristic; the non-finitely-generated surface theorem is real but rests on a sketched key proposition.","tokens_in":13345,"tokens_out":3154,"would_cite":true,"duration_ms":32491,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J28","14J50","14G17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The cone conjecture for Enriques surfaces holds in every odd characteristic.","keywords":["Enriques surfaces","cone conjecture","automorphism groups","positive characteristic","K3 surfaces","supersingular K3 surfaces","crystalline Torelli theorem","effective nef cone"],"falsifier":"For the non-finite-generation example, compute the representation $\\rho$ of the inertia group at $Q_{32}$ on the tangent space of $H_2$ and check whether every element of $\\{t^{-2n}a\\}_{n\\ge 0}$ lies in its image; if even one power is missing, the proof of Theorem 4.3 fails. For the cone theorem, produce a finite-height Enriques surface whose characteristic-0 lift has $W_0 \\rtimes \\operatorname{Aut}(X_0)^*$ of infinite index in $O(L)^+$, which would refute Theorem 3.6(3).","tokens_in":12321,"feed_emoji":"📐","tokens_out":10522,"duration_ms":96743,"temperature":0.7,"pith_summary":"Over an algebraically closed field of odd characteristic, this paper proves the cone conjecture for Enriques surfaces: the automorphism group acts on the effective nef cone with a rational polyhedral fundamental domain. As a direct consequence, the automorphism group is finitely generated, and there are only finitely many smooth rational curves and elliptic fibrations up to automorphism. The proof reduces to the K3 cover, using crystalline Torelli for supersingular covers and characteristic-zero lifting for finite-height covers to place the relevant group as a finite-index subgroup of an arithmetic orthogonal group. The paper also constructs, for p greater than 3, a smooth projective surface birational to an Enriques surface whose automorphism group is discrete but not finitely generated, showing that the cone theorem for the minimal model does not force finite generation after blow-up.","feed_headline":"Enriques surfaces satisfy the cone conjecture in odd characteristic","feed_subtitle":"Automorphisms get a polyhedral fundamental domain on the nef cone, giving finite generation and finiteness of curve classes.","key_machinery":"The load-bearing object is the equivariant Weyl group $W = \\langle R_b \\mid b \\in \\mathcal{N}\\rangle$, where $R_b = r_b \\circ r_{\\theta^*(b)}$ and $\\mathcal{N}$ is the set of nodal classes on the K3 cover orthogonal to their $\\theta$-translate. Lemma 3.5 shows that $W$ preserves the $\\theta$-invariant lattice $L$ and that the effective nef cone $\\mathcal{A}$ is a fundamental domain for $W$ on the positive cone $C^+$. Theorem 3.6 then assembles $G = W \\rtimes \\operatorname{Aut}(X)^*$ and proves $G$ has finite index in the arithmetic group $O(L)^+$, using the crystalline Torelli theorem in the supersingular case and a characteristic-zero lifting in the finite-height case. Finite index in an arithmetic group is what supplies the rational polyhedral fundamental domain and, together with finite generation of arithmetic groups, the finite generation of $\\operatorname{Aut}(X)$.","core_discovery":"The central claim is that every Enriques surface over an algebraically closed field of odd characteristic satisfies the cone conjecture: the automorphism group acts on the effective nef cone with a rational polyhedral fundamental domain. The proof passes to the K3 cover $\\tilde X$ with its fixed-point-free involution $\\theta$, fixes the $\\theta$-invariant sublattice $L$ of the Néron–Severi lattice, and studies the equivariant Weyl group generated by products $r_b \\circ r_{\\theta^*(b)}$ of reflections in orthogonal nodal pairs. The key finiteness step is that $W \\rtimes \\operatorname{Aut}(X)^*$ has finite index in the arithmetic group $O(L)^+$: for supersingular covers this comes from the crystalline Torelli theorem, and for finite-height covers from lifting to characteristic zero. That finite index yields the rational polyhedral fundamental domain and, via finite generation of arithmetic groups, the finite generation of $\\operatorname{Aut}(X)$. The constructive half produces a blow-up of an Enriques surface whose automorphism group is discrete but not finitely generated.","pith_inferences":["The odd-characteristic restriction is essential to the method: characteristic 2 Enriques surfaces need not admit an étale K3 cover, so the two-branch proof would require a different mechanism there.","The blow-up construction shows that finite generation of automorphism groups is not stable under blow-up in positive characteristic, even when the underlying minimal surface satisfies the cone conjecture.","The representation-theoretic mechanism behind the non-finite-generation example, where powers of a transcendental coordinate appear in the image of a local inertia group, looks like a generic source of non-finite generation; similar constructions on other surfaces with a one-parameter local action should yield new examples.","A natural next step would be to make the p=3 analogue explicit using quasi-fibrations, as the paper notes but does not carry out."],"forward_implications":["If the paper is correct, the cone conjecture holds for every Enriques surface over an algebraically closed field of odd characteristic.","The automorphism group of every such Enriques surface is finitely generated, extending the characteristic-zero finiteness statement to odd positive characteristic.","On an Enriques surface in odd characteristic, there are only finitely many smooth rational curves and only finitely many elliptic fibrations up to automorphism.","For any smooth projective surface over a prime field birational to an Enriques surface, base change to any field extension still yields a finitely generated automorphism group.","For p greater than 3 there exists a smooth projective surface birational to an Enriques surface whose automorphism group is discrete but not finitely generated, answering the characteristic-p analogue of a question posed for the complex construction."],"supporting_citations":[{"why":"Proves the cone conjecture for K3 surfaces in odd characteristic and supplies the characteristic-0 lifting and specialization comparison used for finite-height covers.","marker":"[LM18]"},{"why":"Gives the crystalline Torelli theorem for supersingular K3 surfaces, used to identify automorphisms with period-preserving isometries in the supersingular branch.","marker":"[Og83]"},{"why":"Provides the characteristic-zero finite-index statement for the automorphism-lattice group in O(L)^+, on which the finite-height branch of the finite-index proof rests.","marker":"[Do84]"},{"why":"Supplies the quotient-type argument that Section 3 follows to reduce the Enriques statement to its K3 cover.","marker":"[OS01]"},{"why":"Provides the general result that a finite-index subgroup of an arithmetic group acting on a self-dual homogeneous cone admits a rational polyhedral fundamental domain.","marker":"[AMRT10]"},{"why":"Supplies the reflection-group fact that the chamber A is a fundamental domain for W on the Lobachevsky space C^+.","marker":"[Vi71]"},{"why":"Provides the blow-up construction and the detailed argument for Proposition 4.4 that the non-finite-generation example depends on.","marker":"[KO19]"},{"why":"Gives the odd-characteristic surface construction and the lemmas on discreteness and finite generation under base change used in Proposition 1.6.","marker":"[Og19]"},{"why":"Establishes finite generation of arithmetic groups, used to conclude that Aut(X) is finitely generated from finite index in O(L)^+.","marker":"[BH62]"}],"fun_headline_variants":["Enriques surfaces in odd characteristic satisfy the cone conjecture","Automorphisms of Enriques surfaces in odd char get polyhedral fundamental domain","Odd characteristic Enriques surfaces satisfy the cone conjecture","Cone conjecture proven for Enriques surfaces in odd characteristic","Enriques surfaces in odd char: automorphism group finitely generated"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-height case can be lifted to characteristic 0 without losing the finite-index property of the automorphism-lattice group, and that the unproved detail in the non-finite-generation example—that all powers $t^{-2n}a$ really occur in the image of the local inertia representation—is correct.","fun_headline_variants_meta":{"raw":{"variants":["Enriques surfaces in odd characteristic satisfy the cone conjecture","Automorphisms of Enriques surfaces in odd char get polyhedral fundamental domain","Odd characteristic Enriques surfaces satisfy the cone conjecture","Cone conjecture proven for Enriques surfaces in odd characteristic","Enriques surfaces in odd char: automorphism group finitely generated"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":2878,"prompt_tokens":815,"completion_tokens":2063,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":1979}},"tokens_in":431,"tokens_out":2063,"duration_ms":14267,"temperature":1.0,"reasoning_tokens":1979,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:21:08.623610+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the non-finite-generation example, compute the representation $\\rho$ of the inertia group at $Q_{32}$ on the tangent space of $H_2$ and check whether every element of $\\{t^{-2n}a\\}_{n\\ge 0}$ lies in its image; if even one power is missing, the proof of Theorem 4.3 fails. For the cone theorem, produce a finite-height Enriques surface whose characteristic-0 lift has $W_0 \\rtimes \\operatorname{Aut}(X_0)^*$ of infinite index in $O(L)^+$, which would refute Theorem 3.6(3).","supporting_citations":[],"review_version":1}