pith. sign in

arxiv: 2412.13624 · v2 · submitted 2024-12-18 · 🧮 math.AG

On the rationality of some real threefolds

Olivier Benoist , Alena Pirutka This is my paper

Pith reviewed 2026-05-23 07:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords rationalityreal threefoldsconic bundlesquadric surface bundlesunramified cohomologybirational rigidityreal closed fields
0
0 comments X

The pith

Some geometrically rational threefolds over real closed fields are irrational even when intermediate Jacobian obstructions vanish.

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

The paper studies geometrically rational three-dimensional conic and quadric surface bundles over real closed fields whose real locus is connected and whose intermediate Jacobian obstructions to rationality are zero. It produces both examples that fail to be rational, via unramified cohomology and birational rigidity, and examples that are rational, via explicit constructions. A reader would care because rationality over the base field decides whether these varieties admit a parametrization by rational functions, which controls their birational classification. The results separate the two behaviors under the stated conditions on the real locus and obstructions.

Core claim

We obtain both negative and positive results on the rationality of geometrically rational three-dimensional conic and quadric surface bundles defined over real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish, using unramified cohomology and birational rigidity techniques as well as concrete rationality constructions.

What carries the argument

Unramified cohomology and birational rigidity techniques to detect irrationality, paired with explicit parametrizations to detect rationality.

If this is right

  • Certain such bundles are irrational over the real closed field.
  • Certain other such bundles admit explicit rational parametrizations over the real closed field.
  • The separation between rational and irrational cases persists when the base field is any real closed field rather than the reals alone.
  • Unramified cohomology supplies obstructions that survive after the intermediate Jacobian obstructions have been removed.

Where Pith is reading between the lines

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

  • The same techniques could be tested on fourfolds or higher-dimensional bundles with analogous vanishing conditions.
  • The positive rationality constructions may extend to families where the base is a curve of higher genus.
  • The negative results suggest that unramified cohomology remains effective even when classical topological obstructions are absent.

Load-bearing premise

The real locus is connected and the intermediate Jacobian obstructions to rationality vanish for the bundles under study.

What would settle it

An explicit three-dimensional conic or quadric surface bundle over a real closed field with connected real locus, vanishing intermediate Jacobian obstructions, and a rationality status that contradicts the negative or positive result obtained for its class.

read the original abstract

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the rationality of geometrically rational three-dimensional conic and quadric surface bundles defined over the reals and more general real closed fields, assuming the real locus is connected and intermediate Jacobian obstructions vanish. It claims both negative results on non-rationality (via unramified cohomology and birational rigidity techniques) and positive results (via explicit rationality constructions).

Significance. If the central claims hold, the work extends rationality criteria and obstruction techniques to real closed fields for a specific class of threefolds, combining negative results with constructive positive examples. This is a modest but useful contribution to birational geometry over non-algebraically closed base fields, provided the adaptation of unramified cohomology is rigorously justified.

major comments (1)
  1. [Sections presenting the unramified cohomology arguments (likely the core of the negative results)] The negative results rely on unramified cohomology detecting non-rationality for these geometrically rational bundles over real closed fields. Standard unramified cohomology obstructions (e.g., via Brauer group or higher cohomology) are developed over algebraically closed fields; the manuscript must provide an explicit reduction to the algebraic closure or a treatment of the real spectrum/Galois action that preserves the obstruction property. Without this, the applicability to real closed k is not established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment below and are prepared to revise the paper accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Sections presenting the unramified cohomology arguments (likely the core of the negative results)] The negative results rely on unramified cohomology detecting non-rationality for these geometrically rational bundles over real closed fields. Standard unramified cohomology obstructions (e.g., via Brauer group or higher cohomology) are developed over algebraically closed fields; the manuscript must provide an explicit reduction to the algebraic closure or a treatment of the real spectrum/Galois action that preserves the obstruction property. Without this, the applicability to real closed k is not established.

    Authors: We thank the referee for this observation. While the unramified cohomology groups in the paper are defined directly over the real closed base field k (using the standard definition via the function field and discrete valuations), and the non-vanishing is detected on the geometric generic fiber after base change, we agree that an explicit discussion of the compatibility under Galois action and the real spectrum is needed to make the reduction fully rigorous. We will add a short preliminary subsection (approximately one page) that recalls the relevant comparison via the Hochschild-Serre spectral sequence for the Galois cohomology of the algebraic closure and verifies that the obstruction classes remain non-zero when descending to k for our geometrically rational threefolds. This addresses the concern without altering the main arguments. revision: yes

Circularity Check

0 steps flagged

No circularity: standard techniques applied to explicit assumptions

full rationale

The paper applies established methods (unramified cohomology, birational rigidity) to geometrically rational threefolds over real closed fields under stated assumptions (connected real locus, vanishing intermediate Jacobian). No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the abstract or described approach; results are obtained via concrete constructions and obstructions that remain independent of the target claims. The derivation is self-contained against external algebraic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Work rests on standard background assumptions of algebraic geometry over real closed fields; no free parameters, new entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • domain assumption Standard properties of geometrically rational varieties and real closed fields
    Invoked as the setting for the bundles studied.
  • domain assumption Intermediate Jacobian vanishing is a relevant obstruction that can be assumed to disappear
    Paper restricts attention to the case where this obstruction vanishes.

pith-pipeline@v0.9.0 · 5570 in / 1207 out tokens · 30643 ms · 2026-05-23T07:26:50.630511+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write mrnumber output.nonempty.mrnumber newline INTEGERS nameptr namesleft numnames FUNCTION format.language language empty "" " (" language * ")" * if FUNCTION form...

    work page

  2. [2]

    Artin and D

    M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. LMS 25 (1972), 75--95

    work page 1972

  3. [3]

    Ambro, Quasi-log varieties, Tr

    F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), 220--239

    work page 2003

  4. [4]

    Arason, Cohomologische invarianten quadratischer F ormen , J

    J\'on Kr. Arason, Cohomologische invarianten quadratischer F ormen , J. Algebra 36 (1975), no. 3, 448--491

    work page 1975

  5. [5]

    A. A. Avilov, Existence of standard models of conic bundles over algebraically non-closed fields, Sb. Math. 205 (2014), no. 12, 1683--1695

    work page 2014

  6. [6]

    , Birational rigidity of G --del P ezzo threefolds of degree 2 , Mat. Sb. 214 (2023), no. 6, 3--40

    work page 2023

  7. [7]

    Blanc, I

    J. Blanc, I. Cheltsov, A. Duncan and Y. G. Prokhorov, Finite quasisimple groups acting on rationally connected threefolds, Math. Proc. Cambridge Philos. Soc. 174 (2023), no. 3, 531--568

    work page 2023

  8. [8]

    Bochnak, M

    J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), vol. 36, Springer-Verlag, Berlin, 1998, Translated from the 1987 French original, Revised by the authors

    work page 1998

  9. [9]

    Beauville, Vari\'et\'es de P rym et jacobiennes interm\'ediaires , Ann

    A. Beauville, Vari\'et\'es de P rym et jacobiennes interm\'ediaires , Ann. Sci. \' E NS 10 (1977), no. 3, 309--391

    work page 1977

  10. [10]

    34, Cambridge University Press, Cambridge, 1996

    , Complex algebraic surfaces, second ed., LMS Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996

    work page 1996

  11. [11]

    Birkar, Existence of log canonical flips and a special LMMP , Publ

    C. Birkar, Existence of log canonical flips and a special LMMP , Publ. Math. IH\' E S 115 (2012), 325--368

    work page 2012

  12. [12]

    Blanc, S

    J. Blanc, S. Lamy and S. Zimmermann, Quotients of higher-dimensional C remona groups , Acta Math. 226 (2021), no. 2, 211--318

    work page 2021

  13. [13]

    Benoist and O

    O. Benoist and O. Wittenberg, The C lemens- G riffiths method over non-closed fields , Algebr. Geom. 7 (2020), no. 6, 696--721

    work page 2020

  14. [14]

    , Intermediate J acobians and rationality over arbitrary fields , Ann. Sci. \' E NS 56 (2023), no. 4, 1029--1084

    work page 2023

  15. [15]

    Castelnuovo, Sulle superficie di genere 0 , Memorie della Societ \`a dei XL (3) 10 (1896), 103--123

    G. Castelnuovo, Sulle superficie di genere 0 , Memorie della Societ \`a dei XL (3) 10 (1896), 103--123

    work page

  16. [16]

    Clemens and P

    C. Clemens and P. Griffiths, The intermediate J acobian of the cubic threefold , Ann. of Math. (2) 95 (1972), 281--356

    work page 1972

  17. [17]

    Chen, On threefold canonical thresholds, Adv

    J.-J. Chen, On threefold canonical thresholds, Adv. Math. 404 (2022), Paper No. 108447, 1--36

    work page 2022

  18. [18]

    Corti, Factoring birational maps of threefolds after S arkisov , J

    A. Corti, Factoring birational maps of threefolds after S arkisov , J. Algebraic Geom. 4 (1995), no. 2, 223--254

    work page 1995

  19. [19]

    281, Cambridge Univ

    , Singularities of linear systems and 3 -fold birational geometry , Explicit birational geometry of 3-folds, LMS Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259--312

    work page 2000

  20. [20]

    Cheltsov and C

    I. Cheltsov and C. Shramov, Cremona groups and the icosahedron, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016

    work page 2016

  21. [21]

    (N.S.) 25 (2019), no

    , Finite collineation groups and birational rigidity, Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 71, 68

    work page 2019

  22. [22]

    Colliot-Th\'el\`ene and M

    J.-L. Colliot-Th\'el\`ene and M. Ojanguren, Vari\'et\'es unirationnelles non rationnelles: au-del\`a de l'exemple d' A rtin et M umford , Invent. math. 97 (1989), no. 1, 141--158

    work page 1989

  23. [23]

    Colliot- T h\'el\`ene and A

    J.-L. Colliot- T h\'el\`ene and A. Pirutka, Hypersurfaces quartiques de dimension 3: non-rationalit\'e stable , Ann. Sci. \'ENS 49 (2016), no. 2, 371--397

    work page 2016

  24. [24]

    , Certaines fibrations en surfaces quadriques r\'eelles, preprint 2024, arXiv:2406.00463 http://arxiv.org/abs/2406.00463

    work page arXiv 2024

  25. [25]

    Colliot-Th\'el\`ene and A

    J.-L. Colliot-Th\'el\`ene and A. N. Skorobogatov, Groupe de C how des z\'ero-cycles sur les fibr\'es en quadriques , K -Theory 7 (1993), no. 5, 477--500

    work page 1993

  26. [26]

    Cheltsov, Y

    I. Cheltsov, Y. Tschinkel and Zh. Zhang, Rationality of singular cubic threefolds over R , preprint 2024, arXiv:2411.14379 http://arxiv.org/abs/ arXiv:2411.14379

    work page arXiv 2024

  27. [27]

    S. Frei, L. Ji, S. Sankar, B. Viray and I. Vogt, Conic bundle threefolds differing by a constant B rauer class and connections to rationality , preprint 2024, arXiv:2406.13510 http://arxiv.org/abs/2406.13510

    work page arXiv 2024

  28. [28]

    , Curve classes on conic bundle threefolds and applications to rationality, Algebr. Geom. 11 (2024), no. 3, 421--459

    work page 2024

  29. [29]

    Fujino, Fundamental theorems for the log minimal model program, Publ

    O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727--789

    work page 2011

  30. [30]

    Gille and T

    P. Gille and T. Szamuely, Central simple algebras and Galois cohomology , second ed., Camb. Stud. Adv. Math., vol. 165, Cambridge University Press, 2017

    work page 2017

  31. [31]

    C. D. Hacon and J. McKernan, The S arkisov program , J. Algebraic Geom. 22 (2013), no. 2, 389--405

    work page 2013

  32. [32]

    Hassett and Y

    B. Hassett and Y. Tschinkel, Cycle class maps and birational invariants, Comm. Pure Appl. Math. 74 (2021), no. 12, 2675--2698

    work page 2021

  33. [33]

    , Rationality of complete intersections of two quadrics over nonclosed fields, Enseign. Math. 67 (2021), no. 1-2, 1--44

    work page 2021

  34. [34]

    C. D. Hacon and C. Xu, Existence of log canonical closures, Invent. math. 192 (2013), no. 1, 161--195

    work page 2013

  35. [35]

    Iskovskikh and Y

    V. Iskovskikh and Y. Manin, Three-dimensional quartics and counterexamples to the L \"uroth problem , Math. USSR Sbornik 15 (1971), no. 1, 141--166

    work page 1971

  36. [36]

    V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR Izv. 14 (1980), no. 1, 17--39

    work page 1980

  37. [37]

    Ji and M

    L. Ji and M. Ji, Rationality of real conic bundles with quartic discriminant curve, IMRN (2024), no. 1, 115--151

    work page 2024

  38. [38]

    Kawamata, Boundedness of Q - F ano threefolds , Proceedings of the I nternational C onference on A lgebra D edicated to the M emory of A

    Y. Kawamata, Boundedness of Q - F ano threefolds , Proceedings of the I nternational C onference on A lgebra D edicated to the M emory of A . I . M alcev, Contemp. Math., vol. 131, Part 3, Amer. Math. Soc., Providence, RI, 1992, pp. 439--445

    work page 1992

  39. [39]

    Koll\' a r and K

    J. Koll\' a r and K. Matsuki, Termination of canonical flips, Flips and abundance for algebraic threefolds, Ast\'erisque, vol. 211, SMF, Paris, 1992, pp. 157--163

    work page 1992

  40. [40]

    Koll\'ar and S

    J. Koll\'ar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998

    work page 1998

  41. [41]

    Knebusch, On algebraic curves over real closed fields

    M. Knebusch, On algebraic curves over real closed fields. I , Math. Z. 150 (1976), no. 1, 49--70

    work page 1976

  42. [42]

    Koll\'ar, The topology of real algebraic varieties, Current developments in mathematics, 2000, Int

    J. Koll\'ar, The topology of real algebraic varieties, Current developments in mathematics, 2000, Int. Press, Somerville, MA, 2001, pp. 197--231

    work page 2000

  43. [43]

    , Conic bundles that are not birational to numerical C alabi- Y au pairs , \'EPIGA 1 (2017), Art. 1, 14

    work page 2017

  44. [44]

    , Automorphisms and T wisted F orms of R ings of I nvariants , Milan J. Math. 92 (2024), no. 2, 473--499

    work page 2024

  45. [45]

    Kuznetsov and Y

    A. Kuznetsov and Y. Prokhorov, Rationality of F ano threefolds over non-closed fields , Amer. J. Math. 145 (2023), no. 2, 335--411

    work page 2023

  46. [46]

    Manin, Rational surfaces over perfect fields, Publ

    Y. Manin, Rational surfaces over perfect fields, Publ. Math. IH \'E S 30 (1966), 55 --113

    work page 1966

  47. [47]

    Oesinghaus, Conic bundles and iterated root stacks, Eur

    J. Oesinghaus, Conic bundles and iterated root stacks, Eur. J. Math. 5 (2019), no. 2, 518--527

    work page 2019

  48. [48]

    Pfister, Zur D arstellung von -1 als S umme von Q uadraten in einem K \"orper , J

    A. Pfister, Zur D arstellung von -1 als S umme von Q uadraten in einem K \"orper , J. London Math. Soc. 40 (1965), 159--165

    work page 1965

  49. [49]

    Y. G. Prokhorov, The rationality problem for conic bundles, Russian Math. Surveys 73 (2018), no. 3, 375--456

    work page 2018

  50. [50]

    Nauk 76 (2021), no

    , Equivariant minimal model program, Uspekhi Mat. Nauk 76 (2021), no. 3(459), 93--182

    work page 2021

  51. [51]

    413--437

    , Finite groups of birational transformations, European C ongress of M athematics, EMS Press, Berlin, 2023, pp. 413--437

    work page 2023

  52. [52]

    V. G. Sarkisov, Birational automorphisms of conic bundles, Math. USSR Izv. 17 (1981), no. 1, 177--202

    work page 1981

  53. [53]

    USSR Izv

    , On conic bundle structures, Math. USSR Izv. 20 (1982), no. 2, 355--390

    work page 1982

  54. [54]

    Segre, On the rational solutions of homogeneous cubic equations in four variables, Math

    B. Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae 11 (1951), 1--68

    work page 1951

  55. [55]

    V. V. Shokurov, Prym varieties: theory and applications, Math. USSR Izv. 23 (1984), no. 1, 83--147

    work page 1984

  56. [56]

    81, 1996, Algebraic geometry, 4, pp

    , 3 -fold log models , vol. 81, 1996, Algebraic geometry, 4, pp. 2667--2699

    work page 1996