Automorphisms of very general blow up

Supravat Sarkar

REVIEW 5 major objections 4 minor 30 references

Automorphisms of very general blow up

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.07983 v1 pith:UX73KMNC submitted 2026-08-08 math.AG

Supravat Sarkar This is my paper · ORCID

classification math.AG MSC 14J5014J60

keywords automorphismgroupblow-upverygeneralpointselementarytransformationruledsurfacegenerictriviality(-1)-hypersurfaceellipticbundlestructure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\eta^2 = s\cdot\mathrm{id}$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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

Corollary 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.

The 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.

Overall: 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.

Referee Report

5 major / 4 minor

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.

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 (5)

[Section 2, Lemma 2.2] 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 3, Theorem 3.1] 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 3, Corollary 3.2] 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 4, Proposition 4.4] 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 4, Proposition 4.4, parity] 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.

minor comments (4)

[Section 4, Proposition 4.4, statement] 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 4, Proposition 4.6] 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 4, Lemma 4.2] 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 5, Case 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central claim derives from independent dimension-counting and standard external theorems.

full rationale

The paper's main result, Theorem A, is derived from Theorem 3.1, which is proved directly by a dimension-counting argument on the automorphism group scheme: it bounds dim W_{S,\eta,\phi} using the inequality r <= h^0(X,T_X) for n >= 2 and r <= 2h^0(X,T_X) for n = 1. This proof does not assume the conclusion of Theorem A and contains no fitted parameter or normalization that encodes the target result. The remaining cases use standard external facts: the contraction theorem for (-1)-hypersurfaces, [18, Theorem 3.52(2)] for non-uniruled surfaces, and elementary transformations via [23]. The only self-citations, [2] and [3], appear in the introduction as background references and are not load-bearing for the proof. Although Proposition 4.4's proof has an apparent indexing mismatch that may affect its correctness in the ruled-surface case, this is a mathematical gap rather than a circularity: it does not reduce the conclusion to the hypothesis. No step was found where a prediction is equivalent to an input by construction or where a load-bearing premise is justified solely by a self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard theorems: automorphism group schemes are locally of finite type, contraction theorems for (-1)-hypersurfaces, the minimal model theorem for surfaces, classification of uniruled surfaces, and Maruyama's elementary transformation lemma. No free parameters are fitted to data, and no new entities are postulated. The auxiliary constants in the proof, such as r0, c0, and c0', are existence constants rather than fitted parameters.

assumptions (5)

standard math Aut(X) is a group scheme locally of finite type over C, so Aut(X) minus the identity has countably many connected components of dimension h^0(X,T_X).
Used in Theorem 3.1 to form the countable list of non-identity components G_k and to bound the dimension of the fibre g_k^{-1}(phi). This is standard for automorphism schemes of projective varieties.
standard math Existence of a contraction of all (-1)-hypersurfaces on a normal projective variety of dimension at least 3, as stated in Lemma 2.2 and citing [18, Theorem 3.7(3)] and [1, Theorem 3.1].
Used in Case 1 of Theorem A to descend an automorphism of the blow-up to the contracted variety Y.
standard math The birational map from the blown-up surface to its minimal model descends to an isomorphism of a minimal model, via [18, Theorem 3.52(2)].
Used in Case 2 so that psi is an isomorphism and Theorem 3.1 can be applied to the minimal model.
standard math A uniruled nonrational surface admits a ruled surface structure over a nonrational curve, and can be written as a blow-up of a minimal ruled surface at points in distinct fibres.
Used at the start of Case 3 and in Proposition 4.6 to reduce the surface problem to ruled surfaces and elementary transformations.
standard math Elementary transformation lemma: Elm(P(E),p) is isomorphic to (P(F),q) with the expected compatibility of birational maps, as stated in Lemma 2.4 and proved in [23].
Used in Proposition 4.4 to convert an involution of a blow-up into an isomorphism P(E) to P(F) and to produce the matrix eta satisfying eta^2 = lambda id tensor s.

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Cite this review

Pith. "Pith review of Automorphisms of very general blow up." pith.science (2026). https://pith.science/paper/UX73KMNC

@misc{pith2026260807983,
  author       = {Pith},
  title        = {Pith review of: Automorphisms of very general blow up},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UX73KMNC}},
  note         = {Machine review of arXiv:2608.07983}
}

abstract

We show that if $X$ is a projective variety of dimension $\geq 2$ that is not a rational surface, then the blow up of $X$ at sufficiently many very general points has no nontrivial automorphism. Similar results were known before for $\mathbb{P}^2$ and $\mathbb{P}^3.$

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [11]

The pseudoautomorphism group ofP3 blown-up at8very general points

Cécile Gachet. “The pseudoautomorphism group ofP3 blown-up at8very general points”. In:arXiv preprint arXiv:2509.12977(2025)

work page arXiv 2025

[2] [1]

Special rays in the Mori cone of a projective variety

Marco Andreatta and Gianluca Occhetta. “Special rays in the Mori cone of a projective variety”. In:Nagoya Mathematical Journal168 (2002), pp. 127–137

work page 2002

[3] [2]

Automorphisms of punctual Hilbert schemes and symmetric powers of varieties

Ashima Bansal, Supravat Sarkar, and Shivam Vats. “Automorphisms of punctual Hilbert schemes and symmetric powers of varieties”. In:arXiv preprint arXiv:2508.18059(2025)

arXiv 2025

[4] [3]

Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles

Ashima Bansal, Supravat Sarkar, and Shivam Vats. “Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles”. In:Bulletin des Sciences Mathématiques(2026), p. 103795

2026

[5] [4]

Galois groups of random integer polynomials and van der Waerden’s Conjecture

Manjul Bhargava. “Galois groups of random integer polynomials and van der Waerden’s Conjecture”. In:Annals of Mathematics201.2 (2025), pp. 339–377

work page 2025

[6] [5]

The C 1 generic diffeomorphism has trivial centralizer

Christian Bonatti, Sylvain Crovisier, and Amie Wilkinson. “The C 1 generic diffeomorphism has trivial centralizer”. In:Publications Mathématiques de l’IHÉS109 (2009), pp. 185–244

work page 2009

[7] [6]

Rational surfaces with a large group of automorphisms

Serge Cantat and Igor Dolgachev. “Rational surfaces with a large group of automorphisms”. In:Journal of the American Mathematical Society25.3 (2012), pp. 863–905

work page 2012

[8] [7]

Asymptotic laws for knot diagrams

Harrison Chapman. “Asymptotic laws for knot diagrams”. In:Discrete Mathematics & Theoretical Computer Science(2020)

work page 2020