{"id":"86ccec99-26f3-43d3-b032-026482bfd8d0","arxiv_id":"2508.18059","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives an exact classification of surfaces whose punctual Hilbert scheme or symmetric power admits a non-natural automorphism.","lead":"This mathematics paper lists every smooth complex surface whose punctual Hilbert scheme has a symmetry that is not inherited from the surface itself, completing a question posed by Belmans, Oberdieck and Rennemo. It reduces the problem to symmetric powers and produces a short list of exceptional cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's unproved blow-up identification is the load-bearing gap in the Hilbert-to-symmetric reduction.","rationale":"The paper is a substantial contribution and the overall strategy is coherent: reduce the Hilbert-scheme classification to the symmetric-power classification via Theorem B, then classify symmetric-power automorphisms in Theorems C and D. The reader's conditional verdict is appropriate. The most load-bearing point is indeed Theorem 3.2, because without it the equivalence in Theorem B collapses and Theorem A would not follow from Theorem D. My independent reading confirms that the proof of Theorem 3.2 is a sketch at exactly the point where the isomorphism must extend across the exceptional loci: the identification of H\\E1 as a blow-up is cited rather than proved, and the application of Lemma 3.15 is compressed. These gaps are probably fillable, since the cited references and the surrounding lemmas strongly suggest the intended argument, but they are not merely cosmetic: the difference between a blow-up and a merely locally isomorphic P-bundle is what guarantees that base automorphisms lift, and the line-bundle equality in Lemma 3.15 is what guarantees the extension. I do not see a reason to move the verdict to ACCEPT or REJECT; the conditional status is exactly right. The self-citations to [BSV25a] and [BSV25b] reinforce the need for independent verification but are secondary to the Theorem 3.2 gap. My concrete test targets the minimal cases not already covered by the existing literature, and it would settle whether the lift step is sound.","tokens_in":26434,"tokens_out":50176,"duration_ms":551872,"concrete_test":"Check the blow-up claim in the minimal non-covered cases, X=A^3,m=3 and X=A^2,m=4: compute the ideal sheaf of W\\W1 in S^mX and verify that its pullback to H\\E1 is O(-E\\E1) and that the induced map H\\E1 -> Bl_{W\\W1}(S^mX\\W1) is an isomorphism, for instance by comparing exceptional divisors and their normal bundles. If this holds, the lift of φ is immediate from the universal property of blow-ups; then check explicitly that codim_H(E1)≥2 and that the isomorphism φ1^* (p^*φ^*L⊗O(αE)) = p^*L⊗O(αE) holds on H\\E1 with the α obtained from Lemma 3.9. If these checks fail, Theorem 3.2 needs a different proof and Theorem B is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A(1) is derived from Theorem D through Theorem B, and Theorem B is a direct consequence of Theorems 3.1 and 3.2. The lift direction, Theorem 3.2, is the less secure half. Its proof asserts, via [Cha79] and [BK05, Exercise 7.3E(5)], that H\\E1 -> S^mX\\W1 is the blow-up of S^mX\\W1 along W\\W1. This assertion is doing essential work: any automorphism of the base that preserves a smooth centre lifts to a blow-up, but an automorphism of the base need not lift to an arbitrary P-bundle over that centre. One therefore needs to know that the exceptional divisor E\\E1 is the projectivized normal bundle of W\\W1 with the tautological O(-1), and that the map is an isomorphism away from E\\E1. The cited references are not checked for the cases needed, namely m=3 with n≥3 and all m≥4 with n=2. The subsequent application of Lemma 3.15 is also compressed: it requires codim_H(E1) ≥ 2 and an equality of line bundles on U=H\\E1, which is justified only by the sentence 'Since φ1^{-1}(E\\E1)=E\\E1'. This implicitly needs E\\E1 to be a Cartier divisor invariant as a scheme, not merely as a set, and it needs W1=Sing(W) to have the right codimension. None of this is verified in the text. If the blow-up identification is wrong, or if the extension hypotheses fail, then non-natural automorphisms of S^mX need not lift to Hilb^mX, and the classification of Hilbert-scheme automorphisms would not follow from Theorem D.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies, for smooth complex projective surfaces X and integers m≥2, the pairs (m,X) for which the punctual Hilbert scheme Hilb^m(X) admits a non-natural automorphism preserving the big diagonal (Theorem A). The classification has four cases: m=2 with X a product of curves; X an abelian surface isogenous to the square of an elliptic curve; X a simple abelian surface with End_Q(X) not Q and not an imaginary quadratic extension; and X in a class C of isotrivial elliptic surfaces with a non-constant map from the base to a fibre of the Iitaka fibration. The proof reduces the Hilbert-scheme problem to an analogous problem for symmetric powers S^m X (Theorem B), then classifies non-natural automorphisms of symmetric powers of surfaces (Theorem D), with a sufficient criterion in higher dimensions (Theorem C). A corollary treats surfaces of Kodaira dimension at least 1 without the diagonal-preserving hypothesis (Corollary A).","tokens_in":26784,"tokens_out":32559,"duration_ms":280982,"significance":"If correct, Theorem A completely answers Question 1 of Belmans-Oberdieck-Rennemo and provides a full classification of non-natural automorphisms of symmetric powers of surfaces. The reduction Theorem B is a valuable structural bridge between Hilbert schemes and symmetric powers, and Theorem C(i) gives a clean description of Aut(S^2 X) under rigidity hypotheses. The paper contains explicit new examples (the class C surfaces) and the classification statements are precise and falsifiable. The group-theoretic lemmas in Section 4 are well executed. However, the proof of the lift direction Theorem 3.2 is only a sketch with a key step that is not justified, and the descent proof in Theorem 3.1 has notational inconsistencies that obscure the main stratification argument; both are load-bearing for the central result. The paper also depends on two unpublished same-author preprints for structurally important facts.","major_comments":[{"comment":"The proof of the lift direction is not complete. After constructing an automorphism φ1 of U=H\\E1 over the base automorphism φ of S^mX\\W1, the text invokes Lemma 3.15 to extend φ1 to H. Lemma 3.15 requires an equality φ1^*(M|_V)=L|_U of line bundles on U, but the only stated justification is 'Since φ1^{-1}(E\\E1)=E\\E1'. This does not provide the required equality: because φ1 covers φ on the base, φ1^*p^*L = p^*φ^*L, not p^*L, and the two line bundles p^*L⊗O(αE) and p^*φ^*L⊗O(αE) are not shown to agree on U; ρ(U)=1 gives only proportionality, not equality. Moreover, the identification of H\\E1 as the blow-up of S^mX\\W1 along W\\W1 via [Cha79] and [BK05, Exercise 7.3E(5)] is asserted without verifying the needed cases (n=2 with m≥4, and n=3 with m=3). Please supply a complete proof of Theorem 3.2, including a precise verification of the extension criterion.","section":"§3, proof of Theorem 3.2"},{"comment":"The stratification argument in Case 3 uses the notation E_{(t,1)}, E_{(t-1,2,1)}, E_{(2,2,1)}, etc., which is undefined for the stated m because Section 2 defines E_{π,m} only for partitions π of m. For instance, (2,2,1) is not a partition of 4. The intended meaning is clearly the partition with additional trailing 1's, but this shorthand is never introduced, making Lemmas 3.11 and 3.12 as written ill-formed for general m. The induction step 'F_t = E_{(t,1)} for all 3≤t≤m-1' is also inconsistent with F_3 having two components when m=4. Please rewrite the stratification with explicit partitions of m and give a rigorous induction from F_3 to F_m.","section":"§3, Lemmas 3.11-3.12 and Theorem 3.1, Case 3"},{"comment":"The proof lifts an arbitrary automorphism ψ of S^mX to an automorphism φ of X^m by citing [BOR20, Proposition 9 and 12] and asserting that the proof is valid for every smooth projective variety X. Since [BOR20] is stated for surfaces and the branch locus of X^m→S^mX has codimension at least 2 when dim X ≥ 3, the lift is not a formal consequence of the surface argument. This step is load-bearing for Theorem C(i) and hence for Theorem A(2). Please provide a self-contained proof of the lift in the generality used, or restate the required theorem explicitly.","section":"§4, proof of Theorem C(i)"},{"comment":"The paper relies for structurally important facts on two unpublished same-author preprints: [BSV25b, Theorem 2] is used in the proof of Theorem 3.1 (normalization of W and dimension of W_{(m)}), and [BSV25a, Lemma 2.6 and Corollary 4.2] are used in Theorem C(ii) and Remark 3.14. These results are not stated in the manuscript. Please either include the statements, and ideally proofs, of the needed results, or confirm that the preprints are in final accepted form and give precise statements that the reader can verify independently.","section":"§3 and §5, reliance on [BSV25a] and [BSV25b]"}],"minor_comments":[{"comment":"The statement 'ρ(Hilb^m X \\ S^m(X)) = 1' is not meaningful as written, since S^m(X) is not a subset of Hilb^m X; the intended open set is presumably the complement of the exceptional divisor E of the Hilbert-Chow morphism.","section":"§3, Lemma 3.9"},{"comment":"Please introduce explicitly the convention that E_{(a_1,...,a_k)} stands for the stratum associated to the partition of m obtained by adding (m - Σ a_i) ones; this shorthand is used in Lemmas 3.11-3.12 and Theorem 3.1 without definition.","section":"§2 and §3"},{"comment":"The proof of Theorem 5.6 invokes Lemma 5.5, which assumes X is not an abelian surface, but the theorem statement does not include this hypothesis; either add the hypothesis or handle abelian surfaces separately in the proof.","section":"§5, Theorem 5.6"},{"comment":"There is a grammatical typo in the statement: 'LetX is an abelian surface' should be 'Let X be an abelian surface'; similar minor typos occur in Lemma 5.9 and elsewhere.","section":"§5, Lemma 5.2"},{"comment":"The reduction of the fibre of Hilb^3X→S^3X over w∈W_{(3)} to the local model H_{3,n} of Lemma 3.7 is not stated; please clarify the local isomorphism used.","section":"§3, Theorem 3.1 Case 2"},{"comment":"The claim that p^{(m)} is the contraction part of the Albanese map of X^m when p is the contraction part of the Albanese map of X is stated without proof; a brief justification would help the reader.","section":"§5, Theorem 5.6 Step 1"}],"recommendation":"major_revision","confidential_remarks":"The central claims are plausible and the paper contains significant new results, but the proof of Theorem 3.2 is a sketch with a potentially unjustified line-bundle argument, and Theorem 3.1's stratification argument has notational problems that obscure the logic. Both need real work before the paper is acceptable. The editors may also wish to ask the authors to state the results they import from their own unpublished preprints, since the current manuscript is not self-contained on several load-bearing points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague, the headline: this is a real completion of the BOR20 classification, and the symmetric-power classification is new. The paper is worth a serious referee, but you should know that the reduction from Hilbert schemes to symmetric powers has a load-bearing sketch at Theorem 3.2, and the paper leans on two unpublished companion preprints for key facts.\n\nWhat is new and good: Theorem A gives the complete list of surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal: product of curves with m=2, abelian surfaces isogenous to a square of an elliptic curve, simple abelian surfaces with End_Q not Q or imaginary quadratic, and class C surfaces with a non-constant map from base to fibre of the Iitaka fibration. That answers Question 1 of BOR20 and unifies scattered partial results. Theorem D does the same for symmetric powers, which is the right analogue in higher dimension. Theorem C gives useful sufficient conditions in higher dimension. The group-theoretic normalizer computation in Section 4 is careful and convincing. Lemma 3.4, on automorphisms of projective bundles not over the base, is a nice standalone result.\n\nThe soft spots are real. Theorem 3.2 is the bridge: it claims any automorphism of S^mX lifts to Hilb^mX. The proof is a sketch. It invokes Cha79 and BK05 to identify H\\E1 -> S^mX\\W1 as the blow-up of S^mX\\W1 along W\\W1, but does not check the cases needed (m>=3 with n>=3 and m>=4 with n=2), does not check that the exceptional divisor is the projectivized normal bundle with the right O(-1), and the extension argument via Lemma 3.15 is compressed. The line 'Since phi^{-1}(E\\E1)=E\\E1' needs E\\E1 invariant as a scheme, not just as a set, and the codimension of W1 verified. None of that is in the text. If that lift fails, Theorem B collapses, and the Hilbert-scheme classification would not follow from Theorem D. That is a genuine gap, not a nitpick.\n\nAlso, several load-bearing facts are imported from the authors' own unpublished preprints [BSV25a] and [BSV25b]: the normalization of W(3) is X×X, Lemma 2.6 on multiprojective bundles, Corollary 4.2 on symmetric powers determining the variety. The authors are transparent about this, but it does mean the derivation is not self-contained. The reliance is heavy enough that a referee cannot fully verify the paper without those preprints.\n\nThe classification itself may well be right. The descent direction (Theorem 3.1) looks solid, and the surface analysis in Section 5 is systematic. This deserves peer review, not desk rejection. I would ask the authors in revision to expand Theorem 3.2 into a full proof, and to either fold the preprint results into the paper or make them available in complete form. If those two things are addressed, I would take the classification. For a reading group, I would hold off until the companion preprints are out.","headline":"Real completion of the BOR20 classification for surfaces, but the Hilbert-to-symmetric reduction has a genuine gap at Theorem 3.2 and leans on unpublished preprints.","tokens_in":27329,"tokens_out":2929,"would_cite":true,"duration_ms":27332,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J50","14C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies exactly when the Hilbert scheme of points on a surface has an automorphism not induced by the surface, finding four exceptional cases.","keywords":["punctual Hilbert scheme","symmetric power","automorphism group","big diagonal","abelian surfaces","Iitaka fibration","smooth projective surfaces","non-natural automorphisms"],"falsifier":"Find a smooth projective surface $X$ satisfying none of the four conditions of Theorem D and an integer $m\\ge 2$ for which $S^mX$ has a non-natural automorphism; this would directly contradict the claimed classification. More locally, for $X$ a surface with $m=3$, set $W=\\operatorname{Sing}(S^3X)$ and $W_1=\\operatorname{Sing}(W)$, then check whether the automorphism of $H\\setminus E_1$ obtained from the blow-up description extends over $E_1$ by testing whether the two ample line bundles $p^*L\\otimes\\mathcal{O}(\\alpha E)$ and $p^*\\phi^*L\\otimes\\mathcal{O}(\\alpha E)$ agree as sheaves near $E_1$; a mismatch for any surface would break Theorem B.","tokens_in":26152,"feed_emoji":"📐","tokens_out":7262,"duration_ms":66957,"temperature":0.7,"pith_summary":"The paper sets out to classify, for complex smooth projective surfaces, exactly when the punctual Hilbert scheme $\\mathrm{Hilb}^m(X)$ admits an automorphism that is not induced by an automorphism of $X$ but preserves the big diagonal, the locus of non-reduced subschemes. It claims a complete answer: such an automorphism exists precisely in four situations: $m=2$ and $X$ is a product of curves; $X$ is an abelian surface isogenous to the square of an elliptic curve; $X$ is a simple abelian surface whose rational endomorphism algebra is neither $\\mathbb{Q}$ nor an imaginary quadratic field; or $X$ belongs to a specified class of elliptic fibrations with a non-constant map from the base to the fibre. The proof reduces the Hilbert-scheme question to the analogous question for symmetric powers $S^mX$, proves the reduction is an equivalence in the relevant cases, and completes the symmetric-power classification for surfaces. If correct, this closes the classification question raised in [BOR20] and shows that non-natural symmetries are rare and explicitly listable.","feed_headline":"Four surface families produce all exotic Hilbert-scheme automorphisms","feed_subtitle":"For surfaces, non-natural automorphisms of point Hilbert schemes occur only in four explicitly listed cases, closing the open question.","key_machinery":"The load-bearing machinery is the Hilbert-Chow morphism $\\mathrm{Hilb}^m(X)\\to S^mX$, whose exceptional divisor is the big diagonal, paired with a descent-lift equivalence between automorphisms of the Hilbert scheme preserving the big diagonal and automorphisms of the symmetric power. The descent side is carried by a sequence of incidence strata $E_\\pi$: the singular locus of the exceptional divisor forces preservation of successively smaller diagonals until the small diagonal is fixed, and the fibre over the small diagonal has Picard rank one, so curves contracted by Hilbert-Chow are contracted by any such automorphism. The lift side uses the identification of $H$ minus a codimension-two locus with a blow-up of $S^mX$ along its singular locus, together with an extension lemma for isomorphisms of complements of codimension at least two. On the symmetric-power side, the classification is driven by group-theoretic analysis of the normalizer of the symmetric group inside $\\operatorname{Aut}(X^m)$; for $m=2$ the extra swap-within-blocks automorphisms form a group $(\\mathbb{Z}/2\\mathbb{Z})^{k-1}$, and for abelian surfaces the existence of endomorphisms $\\alpha\\neq\\pm 1$ with $m\\mid (1-\\alpha)$ in $\\mathrm{End}(X)$ yields explicit non-natural automorphisms.","core_discovery":"On the paper's own terms, the central discovery is a complete four-part classification. For a smooth projective surface $X$ and integer $m\\ge 2$, the Hilbert scheme $\\mathrm{Hilb}^m(X)$ has an automorphism preserving the big diagonal that is not induced by $\\operatorname{Aut}(X)$ if and only if: (a) $m=2$ and $X$ is a product of two curves; (b) $X$ is an abelian surface isogenous to $E\\times E$ for an elliptic curve $E$; (c) $X$ is a simple abelian surface with $\\mathrm{End}_{\\mathbb{Q}}(X)$ neither $\\mathbb{Q}$ nor an imaginary quadratic extension of $\\mathbb{Q}$; or (d) $X$ lies in class $C$, meaning $X=(D\\times Y)/G$ with $D$ elliptic, $Y$ of general type, and $G$ a finite subgroup of $\\operatorname{Aut}^0(D)$ acting diagonally, and the base of the Iitaka fibration maps non-constantly to the fibre. The paper further claims that the same four conditions characterize non-natural automorphisms of $S^mX$ for surfaces, and that for higher-dimensional varieties with $H^1(X,\\mathcal{O}_X)=0$ or $K_X$ ample, every big-diagonal-preserving automorphism of $\\mathrm{Hilb}^mX$ for $m=2,3$ is natural. The bridge is Theorem B: for $n=2$ or $m\\le 3$, the Hilbert scheme has a non-natural big-diagonal-preserving automorphism if and only if the symmetric power does.","pith_inferences":["The reduction suggests a testable principle for higher dimensions: whenever the relevant Hilbert-Chow fibre has Picard rank one and the exceptional strata form a good stratification, big-diagonal-preserving automorphisms should descend to symmetric powers beyond $m\\le 3$; the current bottleneck is the lift step, whose sketch may require a full blow-up argument.","The class-$C$ examples show that non-natural automorphisms can arise from variation in the fibres of an isotrivial fibration rather than from endomorphism rings of abelian varieties; the same mechanism could plausibly produce exotic automorphisms of symmetric powers of higher-dimensional fibrations.","For abelian varieties of dimension at least two, the paper's remark identifying the kernel of $\\operatorname{Aut}(S^mX)\\to\\operatorname{Aut}(X)$ with a congruence subgroup invites an explicit computation of these kernels for other varieties with large endomorphism rings, such as products of elliptic curves."],"forward_implications":["For surfaces of Kodaira dimension at least one, $\\mathrm{Hilb}^m(X)$ has a non-natural automorphism without any big-diagonal assumption exactly in cases (a) and (d) of Theorem A.","For higher-dimensional varieties with $H^1(X,\\mathcal{O}_X)=0$ or ample canonical bundle, $\\mathrm{Hilb}^2(X)$ and $\\mathrm{Hilb}^3(X)$ admit no non-natural automorphism preserving the big diagonal when $X$ is indecomposable in the $m=2$ case.","The pair consisting of $\\mathrm{Hilb}^mX$ together with its big diagonal determines $X$ up to isomorphism in the cases covered by Theorem B.","For surfaces of Kodaira dimension at least one, the punctual Hilbert scheme $\\mathrm{Hilb}^mX$ alone determines $X$ up to isomorphism.","The classification of non-natural automorphisms of symmetric powers of smooth projective surfaces is now complete and matches the Hilbert-scheme classification."],"supporting_citations":[{"why":"Raises the classification question and supplies the lifting of automorphisms of $S^mX$ to $X^m$, plus the known answers for projective space and surfaces of general type or weak Fano type.","marker":"[BOR20]"},{"why":"Provides the structure of the strata $W_\\pi$ and the identification of the normalization of $W_{(m)}$, used throughout the descent proof and dimensional computations.","marker":"[BSV25b]"},{"why":"Supplies irreducibility and dimensions of the $E_\\pi$ strata for Hilbert schemes of points on surfaces.","marker":"[CM00]"},{"why":"Used in Theorem 3.2 to describe $H\\setminus E_1$ as a blow-up of $S^mX$ along its singular locus.","marker":"[Cha79]"},{"why":"Gives the blow-up identification of the Hilbert-Chow morphism away from the deepest singular locus, a key step in the lift theorem.","marker":"[BK05, Exercise 7.3E(5)]"},{"why":"Supplies the lift of symmetric-power automorphisms in the surface case and the positive answer for abelian surfaces of Picard rank one.","marker":"[Gir24]"},{"why":"Provides the construction of non-natural automorphisms of symmetric powers of abelian surfaces with suitable endomorphism units, generalized in Lemma 5.1.","marker":"[Sas24]"},{"why":"Defines the class of surfaces used in case (d) and describes the connected automorphism groups of elliptic fibrations used in Lemma 5.5.","marker":"[Fon24]"}],"fun_headline_variants":["Four surface families host all exotic Hilbert-scheme automorphisms","Classification complete: Non-natural Hilbert-scheme automorphisms for surfaces","Exotic automorphisms of point Hilbert schemes: four surface cases","Belmans–Oberdieck–Rennemo question answered for surfaces","Non-natural Hilbert-scheme automorphisms classified for surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every automorphism of the symmetric power $S^mX$ lifts to an automorphism of the punctual Hilbert scheme that preserves the big diagonal; the paper's proof of this lift is a sketch that does not fully verify that the isomorphism extends across the locus where the blow-up identification is not literally valid. If that extension fails, the equivalence between Hilbert-scheme automorphisms and symmetric-power automorphisms collapses.","fun_headline_variants_meta":{"raw":{"variants":["Four surface families host all exotic Hilbert-scheme automorphisms","Classification complete: Non-natural Hilbert-scheme automorphisms for surfaces","Exotic automorphisms of point Hilbert schemes: four surface cases","Belmans–Oberdieck–Rennemo question answered for surfaces","Non-natural Hilbert-scheme automorphisms classified for surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000546,"raw_usage":{"total_tokens":2664,"prompt_tokens":1053,"completion_tokens":1611,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":1516}},"tokens_in":669,"tokens_out":1611,"duration_ms":11306,"temperature":1.0,"reasoning_tokens":1516,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:58:29.081971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a smooth projective surface $X$ satisfying none of the four conditions of Theorem D and an integer $m\\ge 2$ for which $S^mX$ has a non-natural automorphism; this would directly contradict the claimed classification. More locally, for $X$ a surface with $m=3$, set $W=\\operatorname{Sing}(S^3X)$ and $W_1=\\operatorname{Sing}(W)$, then check whether the automorphism of $H\\setminus E_1$ obtained from the blow-up description extends over $E_1$ by testing whether the two ample line bundles $p^*L\\otimes\\mathcal{O}(\\alpha E)$ and $p^*\\phi^*L\\otimes\\mathcal{O}(\\alpha E)$ agree as sheaves near $E_1$; a mismatch for any surface would break Theorem B.","supporting_citations":[],"review_version":2}