{"id":"af580f66-9838-42ee-951b-1e9c218456c3","arxiv_id":"2608.07983","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every projective variety of dimension at least two that is not a rational surface, blowing up sufficiently many very general points yields a variety with no nontrivial automorphism.","lead":"This paper proves that blowing up any projective variety of dimension at least two at many very general points destroys all nontrivial automorphisms, provided the variety is not a rational surface. It extends earlier results for the projective plane and projective 3-space to a general setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.4's proof is indexed differently from its statement, so the stated result is not actually proven; the ruled-surface case depends on this proposition.","rationale":"The reader's weakest assumption identified the descent step in the surface case and noted that Proposition 4.4 needs a parity condition. My stress-test finds a more concrete problem: the statement and proof of Proposition 4.4 are indexed inconsistently, so the central inequality of the proof does not follow from the stated hypotheses. This directly affects the ruled-surface case, a load-bearing part of Theorem A. The issue appears repairable by correcting the indexing and stating the parity hypothesis, so I do not move the verdict beyond CONDITIONAL; the reader's conditional verdict remains appropriate. I did not find a flaw in Theorem 3.1 itself, and the descent arguments are standard modulo the under-justified but plausible inequality in Corollary 3.2.","tokens_in":12588,"tokens_out":45397,"duration_ms":372943,"concrete_test":"Re-derive the dimension estimate in Proposition 4.4 from first principles under the statement's indexing: if there are d fixed and r general points (total d+r), the projection of im(κ_{L,s}) to the last r coordinates has dimension at least r, while Lemma 4.2 gives at most deg L+c_0=(d+r)/2+c_0; the resulting bound r≤d+2c_0 does not contradict r_0 for large d. Alternatively, replace the statement with a total of r points (d fixed, r-d general), recompute deg L=r/2, and verify the contradiction; if the proposition is intended in this corrected form, the text must be edited and the even-parity case must be handled explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.4 is stated with points z_1,...,z_{d+r}, with p_1,...,p_d fixed and (p_{d+1},...,p_{d+r}) general, and Y the blow-up at p_1,...,p_{d+r}. The proof, however, uses the indexing of a total of r points: it writes the fibre-product of general coordinates as d<i≤r, derives deg L = r/2, and concludes r-d ≤ dim im(κ_{L,s}) ≤ r/2+c_0, hence r≤2d+2c_0. This inequality is only consistent with the chosen r_0=2(c_0+c_0'+d) if the total number of blown-up points is r, not d+r. With the stated d+r points, the same argument gives deg L=(d+r)/2 and r-d≤(d+r)/2+c_0, i.e. r≤3d+2c_0, which is not contradicted by r≥r_0 when c_0'<d/2. Moreover, the proof requires deg L=r/2 to be an integer, so r must be even; for odd r the argument is vacuous, and this parity condition is never stated. Since Proposition 4.4 is the key input for the ruled-surface case in Proposition 4.6, the proof of Theorem A for nonrational ruled surfaces is incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: if X is a projective variety of dimension at least 2 and X is not a rational surface, then for all sufficiently large r, the blow-up of X at r very general points has trivial automorphism group. The proof combines a dimension-counting statement for automorphisms preserving a finite set up to a fixed closed subset (Theorem 3.1), a lemma on (-1)-hypersurfaces, and a reduction of the non-rational ruled-surface case to elementary transformations of ruled surfaces. An application constructs a smooth projective surface with trivial automorphism group and infinitely many elliptic bundle structures.","tokens_in":12809,"tokens_out":29058,"duration_ms":252198,"significance":"Theorem A is a natural extension of known results for P^2 and P^3, and the elementary-transformation analysis in Section 4 together with the automorphism-group dimension count in Theorem 3.1 are substantive and potentially reusable tools. The application in Section 6 is concise and interesting. If the gaps identified below are repaired, the paper is a solid contribution to the study of generic triviality of automorphism groups under birational transformations.","major_comments":[{"comment":"The proof of Lemma 2.2 contains a false assertion: 'As E_2 is isomorphic to P^{n-1} and f|_{E_2} is nonconstant, f|_{E_2} must be finite.' A nonconstant morphism from P^{n-1} need not have finite fibres; for example, if E_1 and E_2 meet, then f|_{E_2} contracts the positive-dimensional divisor E_1 intersect E_2 to the blown-up point. A correct proof should use that f|_{E_2} is the blow-up of the image divisor at the blown-up point and that E_2 is isomorphic to P^{n-1}; this forces the image to be a P^{n-1} not containing the point, so the two (-1)-hypersurfaces are disjoint. Since Lemma 2.2 underpins the descent argument in Case 1 of Theorem A and the contraction of all (-1)-hypersurfaces, this needs to be fixed.","section":"Section 2, Lemma 2.2"},{"comment":"The proof of Theorem 3.1 argues that if f_k(Gamma_k) is not equal to Z_r for all k, then a very general point avoids the union of the f_k(Gamma_k). This is not justified, because f_k(Gamma_k) is only constructible and may be a proper dense subset of Z_r; such a set is not contained in a countable union of proper closed subvarieties. The argument should instead compare dimensions: if all f_k(Gamma_k) have dimension strictly less than nr, their closures are proper closed subsets and a very general point avoids them; otherwise some k has dim f_k(Gamma_k) = nr, which gives dim Gamma_k at least nr as used later. This is a standard repair, but as written the main dimension-counting lemma has a logical gap.","section":"Section 3, Theorem 3.1"},{"comment":"Corollary 3.2 asserts that r'_0 = h^0(X,T_X) + 1 works because r'_0 is at least dim Aut^0(Y) + 1, where Y is the blow-up of X at m fixed points. The proof does not establish the needed inequality h^0(Y,T_Y) <= h^0(X,T_X). This inequality is true: pushing forward the tangent sheaf of Y gives an injection pi_* T_Y into T_X, so global sections inject. Since Corollary 3.2 is used in the proof of Proposition 4.6, the missing justification should be supplied explicitly.","section":"Section 3, Corollary 3.2"},{"comment":"Proposition 4.4 as stated fixes d+r points, with the last r points general, but the proof is indexed as though the total number of points were r. Specifically, the proof derives deg L = r/2 and uses the inequality r-d <= dim im(kappa_{L,s}), which corresponds to a total of r points and r-d general points. With the stated d+r points one would have deg L = (d+r)/2 and the number of general coordinates is r, giving r <= (d+r)/2 + c_0, i.e. r <= d + 2c_0, which is still contradicted by the chosen r_0. The statement and proof can be reconciled, and the product 'd < i <= r' in the statement should read 'd < i <= d+r', but as written the proof does not prove the stated proposition. Since Proposition 4.4 is the key input for the ruled-surface case in Proposition 4.6, this is load-bearing.","section":"Section 4, Proposition 4.4"},{"comment":"The proof of Proposition 4.4 uses a line bundle L with L^2 = O_C(sum z_i), which forces 2 deg L to equal the number of blown-up points; hence the number of points must be even for the argument to run as written. If the total number of points is odd, the determinant computation shows that the assumed involution moving all exceptional divisors cannot exist, so the proposition is still true, but the proof should state this parity discussion explicitly.","section":"Section 4, Proposition 4.4, parity"}],"minor_comments":[{"comment":"The displayed product in the statement of Proposition 4.4 should be over d < i <= d+r, not d < i <= r; as written, the number of factors does not match the indexing of p_{d+1},...,p_{d+r}.","section":"Section 4, Proposition 4.4, statement"},{"comment":"In the proof of Proposition 4.6, '1 <= j <= p\\'' should almost certainly be '1 <= j <= r\\''; also 'by Lemma 3.1' should refer to Theorem 3.1 or Corollary 3.2, depending on intent. The indexing 'r_0(X_0,s)' in the definition of A should probably be 'r_0(X_0,i)' for 0 <= i <= s.","section":"Section 4, Proposition 4.6"},{"comment":"In the proof of Lemma 4.2, the expression 'dim_eta Z_{L,s}' near the end is a typo for 'dim Z_{L,s}'.","section":"Section 4, Lemma 4.2"},{"comment":"In Case 3, the sentence explaining why an automorphism descends to the base curve should explicitly say that any morphism from a rational curve to a curve of positive genus is constant; this is implicit but worth stating for readability.","section":"Section 5, Case 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious contribution with a plausible overall strategy, and the identified gaps appear local and repairable. The false finiteness claim in Lemma 2.2 and the indexing mismatch in Proposition 4.4 are the most serious issues, but both have evident fixes. I do not see grounds for rejection; a careful revision addressing the listed points should make the proof sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is new and probably right: for any projective variety of dimension at least 2 that is not a rational surface, blowing up at sufficiently many very general points kills all automorphisms. This genuinely generalizes the known P^2 and P^3 statements, and the paper introduces a useful fixed-point criterion (Theorem 3.1) that deserves to be quoted independently.\n\nThe proof is well-structured. Theorem 3.1's dimension count on the automorphism group scheme is clean and gives explicit bounds. The reduction to the ruled-surface case via elementary transformations is the right tool, and the application to a surface with trivial automorphism group and infinitely many elliptic bundle structures is a nice payoff.\n\nThe soft spots are real but repairable. Proposition 4.4, which carries the ruled-surface case, is stated with d+r points but proved with r points: the indices in the statement ('z_{d+r}', 'p_{d+r}', and the product over d<i<=r) are inconsistent, and the proof's degree computation deg L = r/2 only works if the total number of points is r, not d+r. As written, the proposition is not proven. The fix is to correct the statement to have r total points, d fixed and r-d general, or to change the bound accordingly. This is a typo-level error, but it sits in a load-bearing spot, so a referee must ask for it.\n\nCorollary 3.2's proof is one line too short: it asserts r'_0 = h^0(X,T_X)+1 works for any number of blown-up points without showing h^0(Bl_q X, T) <= h^0(X,T). That inequality is true — the differential of the blow-up gives an injection of tangent sections — but it should be stated. Also, the abstract's 'similar results for P^2 and P^3' brushes past the fact that the P^3 result [11] is about pseudoautomorphisms; the introduction is accurate.\n\nThe parity concern in Proposition 4.4 is a non-issue: if the involution preserves no exceptional divisor, the number of divisors it permutes is even, so deg L = r/2 is integral.\n\nOverall: the main theorem is likely correct, the new tools are worth having, and the errors are fixable. The paper deserves a serious referee. I'd suggest sending it out with a request to fix the indexing in 4.4 and to justify Corollary 3.2 properly.","headline":"New and likely correct generalization of the P^2/P^3 blow-up rigidity result, with a clean fixed-point criterion; two fixable but load-bearing indexing and justification gaps in the ruled-surface case.","tokens_in":13366,"tokens_out":21670,"would_cite":true,"duration_ms":172422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J50","14J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every projective variety of dimension at least two that is not a rational surface, blowing up sufficiently many very general points leaves no nontrivial automorphism.","keywords":["automorphism group","blow-up","very general points","elementary transformation","ruled surface","generic triviality","(-1)-hypersurface","elliptic bundle structure"],"falsifier":"Take a concrete non-rational ruled surface such as $X=\\mathbb{P}^1\\times E$ with $E$ an elliptic curve, choose $r$ successively larger, and compute $\\operatorname{Aut}(\\operatorname{Bl}_{p_1,\\dots,p_r}X)$ for a very general $r$-tuple; the paper's proof predicts the group is trivial for all sufficiently large $r$, so any non-identity automorphism found would contradict Theorem A.","tokens_in":12336,"feed_emoji":"💥","tokens_out":15938,"duration_ms":142269,"temperature":0.7,"pith_summary":"The paper proves that enough blow-ups kill all symmetry: if $X$ is a projective variety of dimension at least $2$ and $X$ is not a rational surface, then for all sufficiently large $r$ and very general points $p_1,\\dots,p_r$, the blow-up $\\operatorname{Bl}_{p_1,\\dots,p_r}X$ has trivial automorphism group. This extends earlier results for $\\mathbb{P}^2$ and $\\mathbb{P}^3$ to every non-rational projective variety, and it fits the broad pattern that automorphism groups of suitably generic objects are as small as possible. The proof combines a dimension-counting rigidity lemma for automorphism group schemes with a separate ruled-surface argument using elementary transformations. An application constructs a smooth projective surface with trivial automorphism group but infinitely many distinct elliptic bundle structures.","feed_headline":"Enough point blow-ups leave non-rational varieties without symmetry","feed_subtitle":"A new theorem shows all symmetries disappear after enough very general point blow-ups.","key_machinery":"The load-bearing tool is Theorem 3.1, a rigidity lemma: for a normal projective variety $X$ and a proper closed subset $Y$, once $r>\\dim\\operatorname{Aut}^0(X)+1$ (or $r>2\\dim\\operatorname{Aut}^0(X)+1$ in dimension $1$), no nontrivial automorphism can send a very general $r$-tuple of points into itself together with $Y$. Here 'very general' means outside a countable union of proper subvarieties of $X^r$. For ruled surfaces, the central mechanism is the elementary transformation $\\operatorname{Elm}(X,p)$, which replaces a minimal ruled surface $\\mathbb{P}(E)$ by $\\mathbb{P}(F)$, where $F$ is the kernel of a surjection to skyscraper sheaves; Proposition 4.4 then shows that any involution over the base curve must preserve one of the blown-up fibres. Lemma 4.2 supplies the needed dimension bound for the family of endomorphisms $\\eta$ satisfying $\\eta^2 = s\\cdot\\mathrm{id}$.","core_discovery":"The central claim is Theorem A: if $X$ is a projective variety of dimension at least $2$ and $X$ is not a rational surface, then for all sufficiently large $r$ and very general points $p_1,\\dots,p_r\\in X$, the blow-up of $X$ at these points has no nontrivial automorphism. The proof is divided into three regimes: dimension at least three, non-uniruled surfaces, and uniruled non-rational surfaces. In each regime, an automorphism of the blow-up is forced to descend to an automorphism of a simpler model, and a dimension-counting lemma shows that such a symmetry cannot preserve a very general $r$-tuple once $r$ is large enough.","pith_inferences":["Beyond the paper, the explicit bound $r_0=\\dim\\operatorname{Aut}^0(X)+1$ from Theorem 3.1 suggests that for a fixed variety one could in principle compute the smallest number of points guaranteeing asymmetry, a quantitative question the paper does not pursue.","Beyond the paper, the elementary-transformation machinery may extend from ruled surfaces to higher-rank projective bundles, where the same dimension-counting strategy could kill automorphisms over the base curve or base variety.","Beyond the paper, closing the rational-surface gap would make the statement uniform across all projective surfaces; the known $\\mathbb{P}^2$ case indicates the missing ingredient is a descent argument for automorphisms landing on $\\mathbb{P}^2$ or on Hirzebruch surfaces."],"forward_implications":["Every projective variety of dimension at least three becomes asymmetric after blowing up sufficiently many very general points, because the rational-surface exception is automatic only in dimension two.","For any non-rational projective surface, the blow-up at sufficiently many very general points has trivial automorphism group, so its automorphism group scheme is a single point.","The application gives a smooth projective surface with trivial automorphism group that still admits infinitely many elliptic bundle structures distinct modulo automorphisms, showing that non-minimal surfaces can carry infinitely many such structures.","The remarks show that the 'very general' hypothesis is essential: for abelian varieties, suitably chosen general points can be fixed by a nontrivial automorphism, so the blow-up can retain symmetry."],"supporting_citations":[{"why":"Supplies the contraction theorem used to define and contract $(-1)$-hypersurfaces, and Theorem 3.52(2), which makes a birational self-map of a non-uniruled minimal model an isomorphism.","marker":"[18]"},{"why":"Supplies the contraction theorem used to characterize a $(-1)$-hypersurface as the exceptional divisor of a blow-up at a smooth point.","marker":"[1]"},{"why":"Supplies the elementary-transformation isomorphism $\\operatorname{Elm}(\\mathbb{P}(E),p)\\cong(\\mathbb{P}(F),q)$ that underlies Proposition 4.4.","marker":"[23]"},{"why":"Supplies Exercise II.7.10(d), identifying an isomorphism of projectivized bundles with a vector-bundle isomorphism up to tensoring by a line bundle.","marker":"[14]"},{"why":"Supplies Proposition 1.18, used to contract the $(-1)$-curves lying in fibres of the ruled-surface structure in the uniruled-surface case.","marker":"[15]"}],"fun_headline_variants":["Blow-ups at many points wipe out all symmetries","Non-rational varieties lose symmetries after enough blow-ups","Many blow-ups kill all automorphisms on non-rational varieties","Symmetry vanishes after sufficiently many point blow-ups","Blow up enough points and all symmetries disappear"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For surfaces, the proof assumes that every symmetry of the blown-up surface can be pushed down to a symmetry of a simplified version of the surface (or of its base curve), rather than only to a more general birational map; if that descent fails, the argument breaks.","fun_headline_variants_meta":{"raw":{"variants":["Blow-ups at many points wipe out all symmetries","Non-rational varieties lose symmetries after enough blow-ups","Many blow-ups kill all automorphisms on non-rational varieties","Symmetry vanishes after sufficiently many point blow-ups","Blow up enough points and all symmetries disappear"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000473,"raw_usage":{"total_tokens":2237,"prompt_tokens":722,"completion_tokens":1515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":338,"completion_tokens_details":{"reasoning_tokens":1449}},"tokens_in":338,"tokens_out":1515,"duration_ms":10293,"temperature":1.0,"reasoning_tokens":1449,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:27:59.149009+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete non-rational ruled surface such as $X=\\mathbb{P}^1\\times E$ with $E$ an elliptic curve, choose $r$ successively larger, and compute $\\operatorname{Aut}(\\operatorname{Bl}_{p_1,\\dots,p_r}X)$ for a very general $r$-tuple; the paper's proof predicts the group is trivial for all sufficiently large $r$, so any non-identity automorphism found would contradict Theorem A.","supporting_citations":[{"cited_title":"Kollár and M","cited_arxiv_id":null,"evidence_quote":"Supplies the contraction theorem used to define and contract $(-1)$-hypersurfaces, and Theorem 3.52(2), which makes a birational self-map of a non-uniruled minimal model an isomorphism."},{"cited_title":"Special rays in the Mori cone of a projective variety","cited_arxiv_id":null,"evidence_quote":"Supplies the contraction theorem used to characterize a $(-1)$-hypersurface as the exceptional divisor of a blow-up at a smooth point."},{"cited_title":"Elementary transformations in the theory of algebraic vector bundles","cited_arxiv_id":null,"evidence_quote":"Supplies the elementary-transformation isomorphism $\\operatorname{Elm}(\\mathbb{P}(E),p)\\cong(\\mathbb{P}(F),q)$ that underlies Proposition 4.4."},{"cited_title":"Springer Science & Business Media, 2013","cited_arxiv_id":null,"evidence_quote":"Supplies Exercise II.7.10(d), identifying an isomorphism of projectivized bundles with a vector-bundle isomorphism up to tensoring by a line bundle."},{"cited_title":"Rational surfaces over nonclosed fields","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 1.18, used to contract the $(-1)$-curves lying in fibres of the ruled-surface structure in the uniruled-surface case."}],"review_version":2}