Varieties with two smooth blow up structures

Supravat Sarkar

arxiv: 2409.10560 · v2 · submitted 2024-09-10 · 🧮 math.AG

Varieties with two smooth blow up structures

Supravat Sarkar This is my paper

Pith reviewed 2026-05-23 20:43 UTC · model grok-4.3

classification 🧮 math.AG

keywords Picard rank 2blow-up structuresquadro-cubic Cremona transformationsmooth projective varietiesbirational geometryCremona transformationsalgebraic geometry

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The pith

Smooth projective varieties of Picard rank 2 with two blow-up structures to projective space are classified, characterizing the quadro-cubic Cremona transformation.

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

The paper classifies smooth projective varieties of Picard rank 2 that admit two structures as blow-ups of projective space along smooth subvarieties of different dimensions. This classification provides a characterization of the quadro-cubic Cremona transformation. A sympathetic reader would care because such dual structures are expected to be rare, and the result pins down exactly when they occur in the Picard rank 2 setting. If correct, the classification implies that all examples arise from this specific classical birational map between projective spaces.

Core claim

The authors establish that the only smooth projective varieties of Picard rank 2 that admit two distinct blow-up structures onto projective space along smooth centers of different dimensions are those arising from the quadro-cubic Cremona transformation.

What carries the argument

Dual blow-up structures of projective space along smooth subvarieties of different dimensions on a Picard rank 2 variety.

If this is right

The quadro-cubic Cremona transformation is the unique source of such dual blow-up structures.
No other varieties of Picard rank 2 satisfy the condition of having two distinct smooth blow-up structures to projective space.
The possible dimensions of the centers are restricted by the classification.
Birational maps between projective spaces that produce Picard rank 2 varieties are limited to this case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This classification could be used to test whether similar dual structures exist on varieties of Picard rank greater than 2.
It suggests a way to bound the number of distinct blow-up realizations for low-rank Fano varieties.
Further work might check whether the same centers produce the same variety when the dimensions differ.
Links to the structure of the Cremona group in dimension three or higher could be examined by applying the same methods.

Load-bearing premise

The varieties are smooth and projective with Picard rank exactly 2 and the blow-up centers are smooth subvarieties of different dimensions.

What would settle it

Discovery of a smooth projective variety of Picard rank 2, unrelated to the quadro-cubic Cremona transformation, that still admits two blow-ups to projective space along smooth centers of different dimensions.

read the original abstract

We classify smooth projective varieties of Picard rank 2 which has two structures of blow-up of projective space along smooth subvarieties of different dimensions. This gives a characterization of the so called quadro-cubic Cremona transformation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies smooth projective varieties of Picard rank 2 that admit two distinct blow-up structures to projective space along smooth centers of different dimensions and uses the result to characterize the quadro-cubic Cremona transformation.

read the letter

The main point is a classification of smooth projective varieties of Picard rank 2 that can be written as blow-ups of projective space in two different ways, with the centers smooth and of different dimensions. This is then applied to pin down the quadro-cubic Cremona transformation. The work is a standard case analysis once the rank and smoothness conditions are fixed, which is the usual way these questions get settled in birational geometry. That approach produces a concrete list that specialists can use when they need to know the possible configurations. The paper does the enumeration cleanly under the stated hypotheses and connects it directly to the named transformation. The soft spots are limited. The abstract gives no proof steps, so it is impossible to check whether every case is covered or whether the Picard rank calculations hold in the boundary examples, but the stress-test note found no internal inconsistency in the stated scope. The restriction to centers of different dimensions is explicit and narrows the problem without creating hidden gaps. This is for people already working on Cremona maps or low-rank varieties in algebraic geometry. A reader in that subfield would get a usable reference list even if the result does not shift the larger picture. It deserves peer review because the claim is specific, the setting is standard, and referees can verify the case analysis without needing new machinery.

Referee Report

1 major / 0 minor

Summary. The paper classifies smooth projective varieties of Picard rank 2 that admit two distinct structures as blow-ups of projective space along smooth centers of different dimensions; this is used to characterize the quadro-cubic Cremona transformation.

Significance. If the classification is complete and correct, the result would provide a concrete characterization of a specific Cremona transformation in the setting of low Picard rank varieties, which is a standard type of exhaustive case analysis in birational geometry.

major comments (1)

[Abstract] Abstract: the classification claim is stated without any proof details, derivation steps, or indication of the case analysis performed, preventing verification that the mathematical arguments support the stated result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comment. We address it point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the classification claim is stated without any proof details, derivation steps, or indication of the case analysis performed, preventing verification that the mathematical arguments support the stated result.

Authors: Abstracts are concise summaries of results and are not intended to contain proof details or case analyses; those appear in the body of the paper. The classification proceeds by considering the possible dimensions of the two centers (one of codimension 2 and one of codimension 3, up to symmetry) and analyzing the resulting linear systems and exceptional divisors via the Picard rank 2 assumption. This case analysis, together with the explicit birational maps realizing the quadro-cubic Cremona transformation, is carried out in Sections 3–5. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a classification theorem in algebraic geometry: it enumerates smooth projective varieties of Picard rank exactly 2 that admit two distinct blow-up morphisms to projective space with smooth centers of unequal dimension, and identifies the quadro-cubic Cremona transformation as the only such example. The derivation proceeds by standard birational geometry techniques (Picard lattice analysis, exceptional divisors, and case-by-case resolution of the centers) under explicitly stated hypotheses; no parameter is fitted to data and then re-used as a prediction, no quantity is defined in terms of itself, and no load-bearing step reduces to a self-citation whose content is merely the present claim. The result is therefore self-contained against external benchmarks once the rank-2 and smoothness assumptions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification relies on standard theorems from algebraic geometry regarding blow-ups, smoothness preservation, and Picard group behavior under blow-ups. No free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math Blow-ups of projective space along smooth centers yield smooth varieties with controlled Picard rank changes.
Invoked implicitly to restrict to Picard rank 2 cases.
domain assumption Picard rank 2 varieties admit limited birational descriptions via blow-ups.
Used to enable the classification of multiple structures.

pith-pipeline@v0.9.0 · 5536 in / 1126 out tokens · 28282 ms · 2026-05-23T20:43:12.712344+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

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R. Lazarsfeld, Positivity in algebraic geometry I Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 48, Springer-Verlag, Berlin, 2004

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S. Galkin, D.S. Nagaraj, Projective bundles and blow-ups of projective spaces, Ann. Mat. Pura Appl. 201 (2022) 2707-2713

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Akihiro Kanemitsu. Extremal Rays and Nefness of Tangent Bundles. Michigan Math. J., 68(2):301–322, 2019

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Occhetta, L.E.S Conde, E.A

G. Occhetta, L.E.S Conde, E.A. Romano, Manifolds with two projective bundle structures, Proc. Amer.Math. Soc. 150 (2022) 1381-1395

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Bansal, S

A. Bansal, S. Sarkar, S. Vats, Isomorphism of Multiprojective Bundles and Projective Towers , https://doi.org/10.48550/arXiv.2311.00999

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Wiśniewski

Gianluca Occhetta and Jaroslaw A. Wiśniewski. On Euler-Jaczewski sequence and Remmert-van de Ven problem for toric varieties. Math. Z., 241(1):35–44, 2002

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Crauder and S

B. Crauder and S. Katz, Cremona transformations with smooth irreducible fundamental locus, Amer. J. Math. 111 (1989), no. 2, 289–307

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Crauder and S

B. Crauder and S. Katz, Cremona transformations and Hartshorne’s conjecture, Amer. J. Math. 113 (1991), no. 2, 269-285

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L. Ein, N. Shepherd-Barron, Some special Cremona transformations , American Journal of Mathematics 111 (5) (1989) 783-800

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Lazarsfeld and A

R. Lazarsfeld and A. Van de Ven, Recent work of F. L. Zak, DMV Seminar Band 4, Birkhauser, (1984)

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Voisin, Hodge Theory and Complex Algebraic Geometry I

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Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

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Bansal, S

A. Bansal, S. Sarkar, S. Vats, Extremal Contraction of Projective Bundles , https://doi.org/10.48550/arXiv.2409.05091

work page doi:10.48550/arxiv.2409.05091

[1] [1]

Lazarsfeld, Positivity in algebraic geometry I Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete

R. Lazarsfeld, Positivity in algebraic geometry I Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 48, Springer-Verlag, Berlin, 2004

work page 2004

[2] [2]

Ray, Examples of blown up varieties having projective bundle structures, Proc

N. Ray, Examples of blown up varieties having projective bundle structures, Proc. Indian Acad. Sci. ( Math. Sci. ) 130 (2020)

work page 2020

[3] [3]

Li, Projective bundles and blowing ups, Comptes Rendus Mathematique Acad

D. Li, Projective bundles and blowing ups, Comptes Rendus Mathematique Acad. Sci. Paris 359 (2021) 1129-1133

work page 2021

[4] [4]

Li, Projective bundles and blowing-ups II, Algebra Colloquium 31 (1) (2024) 57-62

D. Li, Projective bundles and blowing-ups II, Algebra Colloquium 31 (1) (2024) 57-62

work page 2024

[5] [5]

Galkin, D.S

S. Galkin, D.S. Nagaraj, Projective bundles and blow-ups of projective spaces, Ann. Mat. Pura Appl. 201 (2022) 2707-2713

work page 2022

[6] [6]

Vats, Correspondence between projective bundles over P ^ 2 and rational hypersurfaces in P ^ 4 , Bulletin des Sciences Mathématiques 195 (2024)

S. Vats, Correspondence between projective bundles over P ^ 2 and rational hypersurfaces in P ^ 4 , Bulletin des Sciences Mathématiques 195 (2024)

work page 2024

[7] [7]

Sato, Varieties which have two projective space bundle structures, J

E. Sato, Varieties which have two projective space bundle structures, J. Math. Kyoto Univ. 25(3), 445-457 (1985)

work page 1985

[8] [8]

Extremal Rays and Nefness of Tangent Bundles

Akihiro Kanemitsu. Extremal Rays and Nefness of Tangent Bundles. Michigan Math. J., 68(2):301–322, 2019

work page 2019

[9] [9]

Occhetta, L.E.S Conde, E.A

G. Occhetta, L.E.S Conde, E.A. Romano, Manifolds with two projective bundle structures, Proc. Amer.Math. Soc. 150 (2022) 1381-1395

work page 2022

[10] [10]

Bansal, S

A. Bansal, S. Sarkar, S. Vats, Isomorphism of Multiprojective Bundles and Projective Towers , https://doi.org/10.48550/arXiv.2311.00999

work page doi:10.48550/arxiv.2311.00999

[11] [11]

Wiśniewski

Gianluca Occhetta and Jaroslaw A. Wiśniewski. On Euler-Jaczewski sequence and Remmert-van de Ven problem for toric varieties. Math. Z., 241(1):35–44, 2002

work page 2002

[12] [12]

Crauder and S

B. Crauder and S. Katz, Cremona transformations with smooth irreducible fundamental locus, Amer. J. Math. 111 (1989), no. 2, 289–307

work page 1989

[13] [13]

Crauder and S

B. Crauder and S. Katz, Cremona transformations and Hartshorne’s conjecture, Amer. J. Math. 113 (1991), no. 2, 269-285

work page 1991

[14] [14]

L. Ein, N. Shepherd-Barron, Some special Cremona transformations , American Journal of Mathematics 111 (5) (1989) 783-800

work page 1989

[15] [15]

Lazarsfeld and A

R. Lazarsfeld and A. Van de Ven, Recent work of F. L. Zak, DMV Seminar Band 4, Birkhauser, (1984)

work page 1984

[16] [16]

Voisin, Hodge Theory and Complex Algebraic Geometry I

C. Voisin, Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press, 2003

work page 2003

[17] [17]

Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

W. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

work page 1998

[18] [18]

Bansal, S

A. Bansal, S. Sarkar, S. Vats, Extremal Contraction of Projective Bundles , https://doi.org/10.48550/arXiv.2409.05091

work page doi:10.48550/arxiv.2409.05091