{"id":"6ffba39d-dbf5-468e-9576-6e7f0dcece78","arxiv_id":"2412.12307","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs explicit non-symplectic involutions on Hilbert squares of K3 surfaces, including a non-natural involution with invariant lattice <2>⊕<-2>.","lead":"This paper builds explicit geometric descriptions of certain symmetry maps (involutions) on a four-dimensional manifold built from a K3 surface. It provides new concrete examples where previously only abstract existence was known.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 39 is conditional on unproved number-theoretic assertions: minimality of (a,b)=(1600k^2-7,5) for Pt(-1) and the no-solution claim for P4t(5); both are asserted in the proof without verification.","rationale":"I read the paper in good faith. Theorem 40 is a genuine geometric construction: the invariant lattice computation via the Beauville conjugation is correct, δ is not invariant, so the involution is non-natural, and the orthogonal complement follows from the isometric embedding via i1^*. The main risk is Theorem 39. The Pell assertions are in fact true and elementary, so the mathematics is likely correct, but as written the proof omits the verification of a needed hypothesis (minimality) and makes an unproved claim (P4t(5) no solution). A referee should request these short arguments before acceptance. Since the reader's verdict is already CONDITIONAL, this stress-test does not change the verdict; it identifies precisely which verification must be supplied.","tokens_in":14134,"tokens_out":36524,"duration_ms":325534,"concrete_test":"Verify the missing hypotheses: for t=(a^2+1)/25 with a=1600k^2-7, prove minimality by noting that any solution with x<a has y<5 and then factoring 25x^2 - a^2 y^2 = y^2-25 as (5x-ya)(5x+ya)=y^2-25<0, impossible since |5x+ya|>25 and the left factor would be a nonzero integer of absolute value less than 1; also check a few small k computationally. Separately, reduce x^2 - 4t y^2 = 5 modulo 8, using t ≡ 2 mod 8, to confirm the P4t(5) no-solution claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 39 (Section 3) applies Proposition 34, whose hypotheses include that (a,b) is the minimal positive solution of Pt(-1): x^2 - t y^2 = -1. For t=1+(64n^2-7)^2 with (a,b)=(64n^2-7,1), minimality is immediate. For the second family, t=(a^2+1)/25 with a=1600k^2-7 and (a,b)=(a,5), minimality is not proved and is not obvious: one must rule out solutions with y<5 and x<a. The text only says 'the only possible value for a is 64n^2-7' and then proceeds. Also, the assertion 'since we can verify that P4t(5) does not have a solution' is made without a proof; although a one-line mod-8 argument works for both t families, the verification is absent. If the proposed (a,b) were not minimal, the divisor b[L_i]-aδ would not be the ample class D_i used in Proposition 34, and the deformation equivalence between (S^[2],σ) and (S_n^[2],κ_i) would not be established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies non-symplectic involutions on the Hilbert square of K3 surfaces, focusing on explicit geometric constructions. For a rank-two Picard lattice generated by H and W with intersection matrix [[4, 8n], [8n, 2]], the author constructs Beauville involutions i1, i2 on S_n^[2] and studies the compositions κ1 = i2 i1 i2 and κ2 = i1 i2 i1. Theorem 39 claims that the unique non-symplectic involution on S^[2] for a 2t-polarized K3 surface deforms into these κi exactly when t belongs to two explicit families. Theorem 40 constructs the non-natural involution ι = i1 φ[2] i1 with invariant lattice ⟨2⟩ ⊕ ⟨-2⟩ and orthogonal complement U^2 ⊕ E8(-1)^2 ⊕ ⟨-2⟩. The paper presents explicit matrix computations and relies on prior classification results; the main issue is a sign error in the formula for D1 and several unproved number-theoretic assertions in the proof of Theorem 39.","tokens_in":14462,"tokens_out":31655,"duration_ms":222112,"significance":"If the sign correction and the missing number-theoretic verifications are supplied, the paper gives genuinely new explicit geometric realizations of non-natural non-symplectic involutions, complementing the existence results of Boissière-Cattaneo-Nieper-Wiesskirchen-Sarti and Boissière-Camere-Sarti. The computation in Theorem 40 is transparent and can be checked by hand, and the construction by conjugating a natural involution with a Beauville involution is elegant. Theorem 39 is conditional on elementary Pell-equation checks that are likely easy to fill; the main obstruction is the sign error and the terse 'only if' part of the proof.","major_comments":[{"comment":"The formula for D1 is incorrect. For n = 1, the stated D1 = 59H + 464W - 57δ has Beauville-Bogomolov square 876034, not 2. The correct class is D1 = -(64n^2-5)H + 8n(64n^2-6)W - (64n^2-7)δ, which has square 2. This error propagates to the definition of L1 in the proof of Theorem 39, where the positive H coefficient is used; with the corrected sign, one obtains L1^2 = 2(1 + (64n^2-7)^2) as required for the first family t = (64n^2-7)^2 + 1. As written, the proof of Theorem 39 is invalid, though the theorem statements may remain true after the correction.","section":"Section 3, Corollary 38"},{"comment":"The assertion 'since we can verify that P_{4t}(5) does not have a solution' is not substantiated. For both claimed families one has t ≡ 2 mod 8, so 4t ≡ 0 mod 8, and the congruence x^2 ≡ 5 mod 8 has no solution. This one-line verification should be included because the absence of solutions to P_{4t}(5) is a hypothesis of Theorem 29 and Proposition 34 and is load-bearing for the existence of the involution σ.","section":"Section 3, proof of Theorem 39"},{"comment":"For the b = 5 family, the minimality of (a,b) = (1600k^2-7, 5) as the minimal positive solution of x^2 - t y^2 = -1 is not proved. The text only says that the only possible value for a is 64n^2-7, but this does not rule out a smaller solution with y < 5. Proposition 34 requires the minimal solution, so a verification must be supplied, for instance by checking the finitely many possible y = 1,2,3,4 via a congruence or descent argument.","section":"Section 3, proof of Theorem 39"},{"comment":"Proposition 5 is stated for d_n = 4(n^2-2) with n a multiple of 4, but Proposition 37 needs the non-existence of solutions to x^2 - 8(8n^2-1)y^2 = -8 for every n > 1. The stated proposition does not apply to this d_n. The proof of Proposition 5 actually works verbatim for d_n = 8(8n^2-1) without a congruence condition on n, so the statement should be corrected; as written, the absence of (-2)-curves, and hence the existence of the Beauville involutions i1 and i2, is not established.","section":"Section 2.1, Proposition 5 and Section 3, Proposition 37"},{"comment":"The 'only if' direction is only sketched. The proof does not explicitly justify that deformation equivalence of (S^[2],σ) with (S_n^[2],κ_i) forces the invariant lattice generator of κ_i to coincide, up to the natural isometry, with b[L_i] - aδ for the minimal solution (a,b) of P_t(-1). Without this identification, the conclusion that the two listed families are the only possible t-values is not fully supported; the argument should spell out how the minimal solution enters and why no other t can occur.","section":"Section 3, Theorem 39"}],"minor_comments":[{"comment":"There are several typographical issues: 'Acknowlegments' should be 'Acknowledgments', 'lenght' should be 'length', and 'sympletic' should be 'symplectic'.","section":"Throughout"},{"comment":"The indices in the phrase 'κi, j=1,2' should read 'κ_i, i=1,2'.","section":"Section 3, Theorem 39"},{"comment":"The second family is printed with ambiguous superscripts; the derivation in the proof indicates it should read t = 2^12 · 5^2 k^4 - 2^7 · 7 k^2 + 2, not the variant with 2^5 that the current rendering suggests.","section":"Section 3, Theorem 39"},{"comment":"The notation '< −2 > ⊕< 2>' should use the standard lattice notation '⟨−2⟩ ⊕ ⟨2⟩'.","section":"Section 4, Theorem 2"},{"comment":"The claim that d_n is not a square and hence there are no nontrivial 0-divisors is true, but it would be helpful to include the short congruence argument (squares mod 64 are not congruent to -8 mod 64).","section":"Section 3, Proposition 37"},{"comment":"The reference [19] is listed with a 2025 volume/year; please update to the final published version if it has appeared.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The sign error in D1 appears to be a typographical slip, but it is load-bearing and must be corrected in the proof of Theorem 39. The number-theoretic gaps are fillable with short arguments, and the 'only if' direction needs to be made explicit. The paper fits the journal's scope and, after revision, should be a reasonable contribution to the explicit geometric realization of automorphisms of hyperkähler manifolds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful paper with one solid new construction and one conditional theorem. Theorem 40 gives an explicit non-natural involution on S_n^[2] with invariant lattice <2>⊕<-2>, by conjugating the natural involution with a Beauville involution. I checked the matrix and the lattice computation; they work. The invariant lattice is spanned by two classes with the right squares and zero mutual pairing, and the orthogonal complement matches the natural-involution case. This distinguishes the example from Ohashi–Wandel and gives people working on automorphisms of Hilbert squares an actual map to compute with. The construction does not assume the target result; it is an explicit composition, so there is no circularity. The reliance on the author's earlier paper [19] is also legitimate: the S_n family is prior published work used as input, not tuned to force the conclusion.\n\nTheorem 39 is a different matter. The idea is sensible: use deformation equivalence to transport the abstract anti-reflection involution to a composition of Beauville involutions on the special surfaces S_n. The deformation framework is in place, and Corollary 38 computes the relevant divisors correctly. But the proof skips two number-theoretic checks that are load-bearing. For the second family, it is asserted that (a,b) = (1600k^2−7,5) is the minimal positive solution of P_t(−1); minimality is not shown and is not obvious, since one must rule out y<5. Separately, the condition that P_{4t}(5) has no solution is asserted with \"since we can verify\" and no verification. A mod-8 argument may indeed handle both stated families, as the stress-test note says, but the text does not give it. These are fixable gaps, not fatal flaws, but right now Theorem 39 is conditional on unstated arithmetic facts.\n\nMinor issue: the theorem statement has \"j=1,2\" while referring to κ_i; the indexing is sloppy but the meaning is clear.\n\nWho is this for? Specialists in hyperkähler automorphisms and K3 surfaces. Theorem 40 is a citable example; Theorem 39 is a reasonable claim backed by a proven deformation framework once the Pell checks are filled in. I would send this to a serious referee rather than desk-reject. The missing arithmetic arguments are exactly the kind of thing a referee can require in revision.","headline":"The explicit Beauville-conjugated example in Theorem 40 is real and worth knowing; Theorem 39 is plausible but rests on two unproved Pell claims that should be supplied.","tokens_in":14919,"tokens_out":1910,"would_cite":true,"duration_ms":18341,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J50","14J28","14J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Non-symplectic involutions on the Hilbert square of a K3 surface are built explicitly by conjugating natural involutions with Beauville involutions.","keywords":["non-symplectic involutions","Hilbert square","K3 surfaces","irreducible holomorphic symplectic manifolds","Beauville involutions","invariant lattices","deformation equivalence","generalized Pell equations"],"falsifier":"Check the two families numerically: find integers $(x,y)$ solving $x^2-4ty^2=5$ for $t=(64n^2-7)^2+1$ or $t=2^{12}\\cdot5^2k^4-2^7\\cdot7k^2+2$ with small $n,k$. One solution would violate the hypothesis on which Theorem 39 rests; a proof of non-solubility, or a verified finite search, would remove the gap.","tokens_in":13933,"feed_emoji":"🔷","tokens_out":16322,"duration_ms":124691,"temperature":0.7,"pith_summary":"The paper's goal is to turn cohomological existence statements about non-symplectic involutions on the Hilbert square $S^{[2]}$ of a K3 surface into maps one can write down and compute. It works with a special family of K3 surfaces whose Picard lattice is generated by two divisors $H,W$ with $H^2=4$, $W^2=2$, and $H\\cdot W=8n$, so their Hilbert squares carry two Beauville involutions. The paper proves that the abstract involution $\\sigma$ deforms to either composition $\\kappa_1=i_2i_1i_2$ or $\\kappa_2=i_1i_2i_1$ through Hilbert squares of K3 surfaces exactly when the polarization half-square $t$ is $1+(64n^2-7)^2$ or the companion quartic family in $k$. It then constructs a non-natural involution $\\iota=i_1\\phi^{[2]}i_1$ with invariant lattice $\\langle2\\rangle\\oplus\\langle-2\\rangle$, giving an explicit geometric model for a classified case that previously had only non-geometric or natural examples. The payoff is that these involutions can be evaluated on subschemes, not just described by their action on $H^2$.","feed_headline":"Hilbert-square involutions get explicit geometric forms","feed_subtitle":"Conjugating natural symmetries by Beauville involutions yields non-symplectic maps and two deformation families.","key_machinery":"Beauville involutions and anti-reflections. A Beauville involution on the Hilbert square of a smooth quartic K3 surface sends a length-2 subscheme to the residual intersection of its line with the quartic, and it acts on $H^2(S^{[2]},\\mathbb{Z})$ as the anti-reflection $v\\mapsto\\langle v,D\\rangle D-v$ in an ample square-2 class $D=[H]-\\delta$. Conjugating any automorphism by a Beauville involution transports invariant lattices by that isometry (Proposition 36). On the auxiliary surfaces $S_n$, the square-2 divisor $W$ gives a natural involution $\\phi^{[2]}$, while $[H]-\\delta$ and $[8nW-H]-\\delta$ give two Beauville involutions $i_1,i_2$; their compositions $\\kappa_1,\\kappa_2$ have rank-1 invariant lattices generated by explicit square-2 classes $D_1,D_2$. The Pell-equation criterion from the existence theorem selects the $t$-values for which the abstract $\\sigma$ has the same divisor data as $D_1$ or $D_2$, making the deformation possible.","core_discovery":"On the paper's own terms, the central discovery is that two lattice-theoretically predicted involutions on Hilbert squares admit explicit geometric models. Theorem 39 states that for $(S,L)$ with $\\operatorname{Pic}S=\\mathbb{Z}L$ and $L^2=2t$, the unique non-symplectic involution $\\sigma$ on $S^{[2]}$ deforms to $\\kappa_1=i_2i_1i_2$ or $\\kappa_2=i_1i_2i_1$ on $S_n^{[2]}$, with every fibre a Hilbert square of a $2t$-polarized K3 surface, exactly when $t=1+(64n^2-7)^2$ or $t=2^{12}\\cdot5^2k^4-2^7\\cdot7k^2+2$. Theorem 40 exhibits $\\iota=i_1\\phi^{[2]}i_1$ as a non-natural non-symplectic involution whose invariant lattice is spanned by $8nH-W-8n\\delta$ and $2H-3\\delta$, therefore isometric to $\\langle2\\rangle\\oplus\\langle-2\\rangle$, with orthogonal complement $U^{\\oplus2}\\oplus E_8(-1)^{\\oplus2}\\oplus\\langle-2\\rangle$.","pith_inferences":["Editorial inference: the same conjugation recipe could produce geometric realizations of other classified non-symplectic automorphisms whenever an ample square-2 divisor and a Beauville involution coexist on a Hilbert square.","Editorial inference: the explicit invariant classes make a computation of the fixed locus of $\\iota$ feasible; a description of that fixed locus would give an independent check of the realization.","Editorial inference: the two $t$-families are singled out by divisibility of the coefficients in $D_1,D_2$; analogous divisibility conditions for larger $b$ may yield further deformation families beyond the two stated.","Editorial inference: substituting other natural involutions for $\\phi^{[2]}$ in the conjugation $i_1(\\cdot)i_1$ would produce a family of non-natural involutions whose invariant lattices are the Beauville-images of the starting natural invariant lattices."],"forward_implications":["For the two families of $t$ in Theorem 39, the abstract non-symplectic involution on $S^{[2]}$ is realized as a composition of two Beauville involutions on $S_n^{[2]}$, so it can be evaluated geometrically rather than only cohomologically.","The deformation connecting $(S^{[2]},\\sigma)$ to $(S_n^{[2]},\\kappa_i)$ can be chosen so every fibre is $\\Sigma^{[2]}$ for a $2t$-polarized K3 surface $\\Sigma$, so the model remains inside Hilbert squares throughout.","The involution $\\iota=i_1\\phi^{[2]}i_1$ is a concrete non-natural involution with invariant lattice $\\langle2\\rangle\\oplus\\langle-2\\rangle$ and the natural-embedding orthogonal complement, providing a geometric representative of that lattice-theoretic case.","The explicit generators $8nH-W-8n\\delta$ and $2H-3\\delta$ of the invariant lattice make the action of $\\iota^*$ on $\\operatorname{NS}(S_n^{[2]})$ explicit through the displayed $3\\times3$ matrix."],"supporting_citations":[{"why":"Supplies Theorem 29: the Pell-condition criterion that guarantees existence and uniqueness of the non-symplectic involution $\\sigma$ on the Hilbert square, together with the divisor $D=b[L]-a\\delta$ from Remark 30.","marker":"[6]"},{"why":"Constructs the auxiliary K3 surfaces $S_n$ with Picard lattice generated by $H,W$ and the square-2 divisor $W$ inducing the involution $\\phi$; these are the geometric stage for both theorems.","marker":"[19]"},{"why":"Supplies Theorem 28 (ample $\\langle2\\rangle$-polarized divisors define anti-reflection involutions) and Theorem 33 (deformation equivalence of such involutions), the deformation machinery used in Theorem 39.","marker":"[5]"},{"why":"Classifies non-symplectic automorphisms of K3-type fourfolds by $p$-elementary lattices and determines primitive embeddings of $\\langle2\\rangle\\oplus\\langle-2\\rangle$, identifying the orthogonal complement of $\\iota$.","marker":"[7]"},{"why":"Criterion that an involution of $S^{[2]}$ is natural iff it preserves the exceptional divisor; used to certify that $\\iota$ is non-natural.","marker":"[9]"},{"why":"Beauville's structure theorem for $H^2(S^{[n]},Z)=H^2(S,Z)\\oplus Z\\delta$ and the Beauville-Bogomolov form, the lattice framework for invariant lattices.","marker":"[2]"},{"why":"Introduces Beauville involutions on Hilbert squares of quartic surfaces and their reflection action, the basic building block of $\\kappa_1,\\kappa_2,\\iota$.","marker":"[1]"},{"why":"Provides the previously known example of a non-natural involution with invariant lattice $\\langle2\\rangle\\oplus\\langle-2\\rangle$ against which the new geometric realization is compared.","marker":"[17]"}],"fun_headline_variants":["Explicit geometric forms for non-symplectic involutions on Hilbert squares","K3 Hilbert square involutions: concrete models from lattice theory","Non-symplectic involutions on S^[2] get geometric constructions","Lattice-predicted involutions on Hilbert squares become explicit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is the unproved assertion that the generalized Pell equation $x^2-4ty^2=5$ has no integer solution for $t=1+(64n^2-7)^2$ and for $t=2^{12}\\cdot5^2k^4-2^7\\cdot7k^2+2$; Theorem 39 needs this to invoke the existence and uniqueness of $\\sigma$.","fun_headline_variants_meta":{"raw":{"variants":["Explicit geometric forms for non-symplectic involutions on Hilbert squares","K3 Hilbert square involutions: concrete models from lattice theory","Non-symplectic involutions on S^[2] get geometric constructions","Lattice-predicted involutions on Hilbert squares become explicit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1912,"prompt_tokens":870,"completion_tokens":1042,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":964}},"tokens_in":486,"tokens_out":1042,"duration_ms":8726,"temperature":1.0,"reasoning_tokens":964,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:15:15.205104+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the two families numerically: find integers $(x,y)$ solving $x^2-4ty^2=5$ for $t=(64n^2-7)^2+1$ or $t=2^{12}\\cdot5^2k^4-2^7\\cdot7k^2+2$ with small $n,k$. One solution would violate the hypothesis on which Theorem 39 rests; a proof of non-solubility, or a verified finite search, would remove the gap.","supporting_citations":[{"cited_title":"Boissiére, A","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 29: the Pell-condition criterion that guarantees existence and uniqueness of the non-symplectic involution $\\sigma$ on the Hilbert square, together with the divisor $D=b[L]-a\\delta$ from Remark 30."},{"cited_title":"Paiva and A","cited_arxiv_id":null,"evidence_quote":"Constructs the auxiliary K3 surfaces $S_n$ with Picard lattice generated by $H,W$ and the square-2 divisor $W$ inducing the involution $\\phi$; these are the geometric stage for both theorems."},{"cited_title":"Boissiére, A","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 28 (ample $\\langle2\\rangle$-polarized divisors define anti-reflection involutions) and Theorem 33 (deformation equivalence of such involutions), the deformation machinery used in Theorem 39."},{"cited_title":"Boissière, C","cited_arxiv_id":null,"evidence_quote":"Classifies non-symplectic automorphisms of K3-type fourfolds by $p$-elementary lattices and determines primitive embeddings of $\\langle2\\rangle\\oplus\\langle-2\\rangle$, identifying the orthogonal complement of $\\iota$."},{"cited_title":"Boissière and A","cited_arxiv_id":null,"evidence_quote":"Criterion that an involution of $S^{[2]}$ is natural iff it preserves the exceptional divisor; used to certify that $\\iota$ is non-natural."},{"cited_title":"Beauville","cited_arxiv_id":null,"evidence_quote":"Beauville's structure theorem for $H^2(S^{[n]},Z)=H^2(S,Z)\\oplus Z\\delta$ and the Beauville-Bogomolov form, the lattice framework for invariant lattices."},{"cited_title":"Beauville","cited_arxiv_id":null,"evidence_quote":"Introduces Beauville involutions on Hilbert squares of quartic surfaces and their reflection action, the basic building block of $\\kappa_1,\\kappa_2,\\iota$."},{"cited_title":"Non-natural non-symplectic involutions on symplectic manifolds of K3^{[2]}-type","cited_arxiv_id":"1305.6353","evidence_quote":"Provides the previously known example of a non-natural involution with invariant lattice $\\langle2\\rangle\\oplus\\langle-2\\rangle$ against which the new geometric realization is compared."}],"review_version":1}