Pith. sign in

REVIEW 5 minor 40 references

Vari\'et\'es r\'eelles connexes non stablement rationnelles

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper constructs the first smooth intersections of two quadrics in $\mathbb{P}^5$ and smooth cubic hypersurfaces in $\mathbb{P}^4$ over the field of real Puiseux series that are not stably rational even though their real point spaces…

desk verdict A strong paper: first stable non-rationality examples with connected real locus over real Puiseux series, built on a clean semialgebraic connectedness criterion and a flexible specialization method. read the letter →

arxiv 2505.21477 v2 pith:JWR73H4L submitted 2025-05-27 math.AG

Jean-Louis Colliot-Thélène , Alena Pirutka , Federico Scavia This is my paper
classification math.AG MSC 14E0814M2014P1014P2514F20
keywords Rationalityrealconnectednesssemi-algebraicgeometryspecialisationunramifiedcohomologyquadraticformsChowgroupsPuiseuxseries
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

Over the field of real Puiseux series $R\{\{t\}\}$, the paper constructs the first smooth intersections of two quadrics in $\mathbb{P}^5_R$ and smooth cubic hypersurfaces in $\mathbb{P}^4_R$ that are not stably rational, while the space $X(R)$ of their $R$-points is semi-algebraically connected. Stable rationality over a real closed field forces the real locus to be connected, so the question was whether connectedness is sufficient; these examples show it is not over the Puiseux reals. The construction deforms singular real models whose real points are connected but whose unramified Brauer group is nontrivial, and a specialization argument turns that Brauer class into a nonzero class in the Chow group of zero-cycles of the smooth Puiseux fiber, excluding stable rationality. The same template also produces smooth conic bundles over $\mathbb{P}^2_R$ and higher-dimensional quadric fibrations and intersections of two quadrics with the same combination of properties, while the analogous question over the ordinary real numbers remains open.

What carries the argument

The load-bearing mechanism is a two-step transfer from a singular real model to a smooth Puiseux variety. The connectedness step states that for a projective flat family over $\mathbb{A}^1_R$ with smooth generic fiber, if the special fiber's real points are smooth and connected, then the real Puiseux points of the generic fiber are semi-algebraically connected; the proof uses a Nash triviality theorem for proper semi-algebraic submersions or a real etale base change theorem. The irrationality step is a specialization method for zero-cycles: given a geometric Brauer obstruction on the singular fiber, one obtains a nonzero class in $A_0$ of the generic fiber of a deformation, provided the relevant norm homomorphism on Chow groups is an isomorphism, a condition that holds when the singular locus is finite and defined over $\mathbb{C}$, and more generally under a fiberwise $\mathrm{CH}_0$-triviality hypothesis. The Brauer class is supplied by residue computations on explicit singular models, and in higher dimension by degree-three unramified cohomology classes detected through injectivity results for quadratic forms.

What would settle it

Choose explicit $q_1,q_2$ in the two-quadric deformation of Theorem 3.6 with $\det(q_1+tq_2)$ separable, and compute the specialization map on zero-cycles from the Puiseux generic fiber to the singular fiber $Y$; if the difference between the generic point and a constant $R$-point maps to zero in $\mathrm{CH}_0(Y_{R(Y)})$, then $X$ is $\mathrm{CH}_0$-trivial and the theorem's conclusion fails for that example.

Watch

Extended reading notes

Core claim

The paper's central claim is that smooth complete intersections of two quadrics in $\mathbb{P}^5_{R\{\{t\}\}}$ and smooth cubic hypersurfaces in $\mathbb{P}^4_{R\{\{t\}\}}$ constructed as the Puiseux fibers of deformations of suitable singular real models are not $\mathrm{CH}_0$-trivial, hence not stably rational, although their real point sets are semi-algebraically connected. The proof has two halves. A semi-algebraic triviality theorem for proper smooth families shows that when the special fiber's real points are smooth and connected, the Puiseux fiber's real points are semi-algebraically connected, via either Nash triviality or real etale base change. A specialization theorem transfers a nonzero unramified Brauer class on the singular special fiber into a nonzero class in $A_0(X\times_{R\{\{t\}\}}K(X))$, using a standard pairing between zero-cycles and unramified cohomology; this rules out $\mathrm{CH}_0$-triviality. Higher-dimensional analogues replace the Brauer group by degree-three unramified cohomology and require an explicit resolution of singularities.

Load-bearing premise

The construction depends on the singular special fiber $Y$ possessing an unramified Brauer class that does not come from the base field and that survives the deformation to the generic fiber; if that class vanished, the examples could be $\mathrm{CH}_0$-trivial even though their real loci are connected.

Editorial extensions

If this is right

  • Semi-algebraic connectedness of the real locus does not imply stable rationality over the real Puiseux series field, even for smooth intersections of two quadrics in $\mathbb{P}^5$ and smooth cubic threefolds in $\mathbb{P}^4$.
  • The constructed varieties fail to be retractively rational, not just stably rational, because $\mathrm{CH}_0$-triviality is necessary for both.
  • The same deformation method yields smooth quadric-surface fibrations over $\mathbb{P}^1$ with integral geometric fibers, geometrically rational conic bundles over $\mathbb{P}^2$, and higher-dimensional quadric fibrations and intersections of two quadrics in $\mathbb{P}^9$ with connected real loci and non-$\mathrm{CH}_0$-triviality.
  • Several of the smooth examples have low-degree unramified cohomology equal to that of the base field, so those invariants alone cannot detect the failure of stable rationality; the Chow group of zero-cycles is the discriminating invariant.
  • The question over the ordinary real numbers $\mathbb{R}$ remains open.

Reading between the lines

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

  • A natural next test is whether the same template can be run over $\mathbb{R}$ itself: one would need a one-parameter real family whose generic real fibers are all connected, rather than the infinitesimal Puiseux parameter, together with a singular fiber carrying a nonconstant unramified Brauer class.
  • The connectedness criterion is independent of the rationality content and could be reused: any smooth deformation of a singular real variety with smooth connected real locus will have semi-algebraically connected real Puiseux points, so the main task in producing further non-$\mathrm{CH}_0$-trivial examples is finding singular models with sufficiently rich unramified cohomology.
  • The invariant computations in the paper suggest that for quadric fibrations of relative dimension at least four, real connectedness often forces low-degree unramified cohomology to be constant; this points to zero-cycle specialization as the discriminating invariant for separating real topology from rationality over real closed fields.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper develops a deformation–specialization method to construct smooth projective varieties over the real Puiseux series field R{{t}} whose real point space is semi-algebraically connected but which are not CH0-trivial, hence not stably rational. Starting from a singular R-variety Y whose real points are smooth and connected, whose singularities are non-real, and whose unramified Brauer group contains a nonconstant class vanishing over C, the authors deform Y to a smooth family over A^1_R. Theorem 1.1, proved both via Nash triviality of Coste–Shiota and via Scheiderer's real étale cohomology, shows that the generic fiber over R{{t}} has the same number of semi-algebraically connected components as Y(R). Theorem 3.1 combines this with a specialization argument in the style of Colliot-Thélène–Pirutka to prove non-CH0-triviality of the generic fiber. The method is applied to explicit singular models to produce: smooth intersections of two quadrics in P^5_{R{{t}}} (Theorem 3.6), smooth cubic hypersurfaces in P^4_{R{{t}}} (Theorem 3.9), geometrically rational conic bundles over P^2_{R{{t}}} (Theorem 3.12), and higher-dimensional examples, including relative quadric fibrations of dimension 7 over P^1 and intersections of two quadrics in P^9 (Theorems 5.3 and 5.6). Section 6 computes low-degree unramified cohomology of the examples, showing that in many cases these invariants cannot detect the CH0-nontriviality.

Significance. If the arguments are correct, the paper settles a natural question by providing the first examples over R{{t}} of smooth intersections of two quadrics and smooth cubic hypersurfaces whose real locus is semi-algebraically connected but which are not stably rational; the analogous question over R remains open. The method is original and flexible: it combines a semi-algebraic Ehresmann-type triviality theorem with a specialization argument that does not require an explicit resolution of the singular special fiber in the main low-dimensional cases. The higher-dimensional extensions via H^3 and unramified cohomology, as well as the detailed invariant computations in Section 6, are substantial contributions. The paper is carefully written, with explicit equations for the singular models and detailed verification of most hypotheses; the main argument does not assume the target result and relies on independent published theorems.

minor comments (5)
  1. [§3.1–3.3, Lemmas 3.2(d), 3.5(d), 3.8(d)] Condition (d), the existence of a nonconstant unramified Brauer class on the singular model, is the load-bearing input for non-CH0-triviality, but it is imported entirely from [CTP24, Prop. 11.1]. I have no reason to doubt that proposition, but since Theorem 3.1(ii) and all main theorems depend on it, the paper would be more robust if the authors either stated the exact assertion of [CTP24, Prop. 11.1] or added a short direct residue computation for the quaternion class (−1,u) on the explicit equation of §3.1 and its birational transfers.
  2. [§3.4, Lemma 3.11(d)] The claim that Ker[H^2(R(P^2),Z/2)→H^2(R(Z),Z/2)] is generated by (−1,p1p2), and the verification that α=(−1,p1) is not in this kernel, are compressed into the phrase "par un calcul de résidus". Since this is essential for Theorem 3.12, please expand the calculation or give a precise reference for the kernel statement.
  3. [§1.1, first proof of Theorem 1.1] The sentence "On a donc V(R)=X(R)" appears to mix R-points and R-points; the intended statement seems to be that every R-point of X lies in the smooth locus V of f. Please clarify the notation in that paragraph.
  4. [Theorem 5.6] The statement contains a duplicated word: "dans dans P^9" should read "dans P^9".
  5. [Introduction, comparison with [HT21]] The introduction says that Hassett–Tschinkel and Benoist–Wittenberg establish "non-rationalité" of intersections of two quadrics, while the paper's goal is stable non-rationality; it would be helpful to state explicitly that those earlier results do not settle stable rationality, since the distinction is central to the paper's contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction uses prior theorems as independent inputs, and no claim reduces by definition or by fit to its own assumptions.

full rationale

The derivation chain is self-contained in the required sense. Semi-algebraic connectedness of X(R{{t}}) is obtained from Theorem 1.1, which is proved via the Coste-Shiota Nash triviality theorem and via Scheiderer's real etale cohomology; neither proof assumes the target non-stable-rationality result. Non-CH0-triviality is obtained from Theorem 2.3, a specialization argument adapted from [CTP16], using the Merkurjev pairing and two explicit hypotheses: injectivity of the map Phi (Lemma 2.4) and non-surjectivity of the map on unramified cohomology (Theorem 3.1(d)). The latter is verified in Lemmas 3.2(d), 3.5(d) and 3.8(d) by citing [CTP24, Proposition 11.1] for the quaternion class (-1,u). This is a genuine citation to a prior theorem, not a restatement of the present claim; the cited proposition is parameter-free and its stated assumptions do not include the conclusion that the new smooth X are non-stably-rational. There are no fitted parameters renamed as predictions, no quantity defined in terms of the quantity it is said to predict, and no uniqueness theorem imported to force the construction. The dependence on [CTP24] is a verification dependency, not a circular one: whether that cited proposition is correct is a correctness-risk question, not evidence that the derivation assumes its own conclusion. Accordingly, no specific reduction of an output to an input can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on a standard toolbox of algebraic geometry and quadratic form theory; no numerical parameters are fitted and no new entities are postulated. The paper's new input is the semialgebraic connectedness criterion (Theorem 1.1) and the adapted specialization argument. The auxiliary singular varieties and their Brauer classes come from prior published work, notably [CTP24]. The arbitrary polynomial P(u) and generic quadratic forms q1,q2 are construction choices, not fitted parameters.

assumptions (7)
  • standard math Hironaka resolution of singularities (characteristic zero).
    Used in Lemma 1.4 and Situation 2.1 to produce a smooth projective birational model Z of the singular Y with p^{-1}(U) isomorphic to U.
  • domain assumption Coste-Shiota Nash triviality theorem for proper Nash submersions.
    First proof of Theorem 1.1; requires the family to be a proper Nash submersion over an interval, which is verified via the Artin-Lang homomorphism theorem.
  • domain assumption Scheiderer's real etale cohomology and proper base change.
    Second proof of Theorem 1.1 and proof of Theorem 1.3; supplies the constancy of semialgebraic component counts under specialization.
  • standard math Merkurjev's norm residue theorem (degree 2).
    Used in Section 6 to relate H^3(k,Z/2) and H^3(k,Q/Z(2)) and to prove unramified cohomology vanishing results.
  • standard math Kahn-Rost-Sujatha Theorem 5 on unramified cohomology of quadrics.
    Used repeatedly in Section 6 (Propositions 6.6, 6.11, 6.13, 6.18) to compute H^3_nr for quadric fibrations.
  • standard math Specialization method of Colliot-Thelene-Pirutka ([CTP16, Prop 1.4, Thm 1.12]).
    Gives the bridge from CH0-triviality over R{{t}} to CH0-triviality over R((t^{1/n})) and the specialization map used in Theorem 2.3.
  • standard math Arason's theorem on H^3-injectivity for non-Pfister quadrics.
    Used in Lemma 5.1 to produce an unramified H^3 class on the dimension-6 quadric fibration.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Vari\'et\'es r\'eelles connexes non stablement rationnelles." pith.science (2026). https://pith.science/paper/JWR73H4L

@misc{pith2026250521477,
  author       = {Pith},
  title        = {Pith review of: Vari\'et\'es r\'eelles connexes non stablement rationnelles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWR73H4L}},
  note         = {Machine review of arXiv:2505.21477}
}
abstract

Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably rational but for which the space $X(R)$ of $R$-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers $\mathbb{R}$ remains open.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

40 extracted references · 35 canonical work pages

  1. [1]

    J\'on Kr. Arason. Cohomologische Invarianten quadratischer F ormen. J. Algebra , 36(3):448--491, 1975

    work page 1975

  2. [2]

    Artin, A

    M. Artin, A. Grothendieck et J. L. Verdier (eds.). Th\' e orie des topos et cohomologie \' e tale des sch\' e mas. Tome 2 . Lecture Notes in Mathematics, Vol. 270. Springer-Verlag, Berlin–New York, 1972. S\' e minaire de G\' e om\' e trie Alg\' e brique du Bois-Marie 1963--1964 (SGA 4), avec contributions par N. Bourbaki, P. Deligne et B. Saint-Donat

    work page 1972

  3. [3]

    Universal unramified cohomology of cubic fourfolds containing a plane

    Asher Auel, Jean-Louis Colliot-Th\' e l\`ene et Raman Parimala. Universal unramified cohomology of cubic fourfolds containing a plane. Brauer groups and obstruction problems , volume 320 de Progr. Math. , pages 29--55. Birkh\" a user/Springer, Cham, 2017

    work page 2017

  4. [4]

    On the rationality of some real threefolds

    Olivier Benoist et Alena Pirutka. On the rationality of some real threefolds. arXiv:2412.13624 , 2024

    arXiv 2024

  5. [5]

    Intermediate Jacobians and rationality over arbitrary fields, 2019

    Olivier Benoist et Olivier Wittenberg. Intermediate Jacobians and rationality over arbitrary fields, 2019. Ann. Sci. \'Ec. Norm. Sup\'er. (4) 56(4):1029--1084, 2023

    work page 2019

  6. [6]

    On the integral Hodge conjecture for real varieties, I

    Olivier Benoist et Olivier Wittenberg. On the integral Hodge conjecture for real varieties, I. Invent. Math. , 222(1):1--77, 2020

    work page 2020

  7. [7]

    On the integral Hodge conjecture for real varieties, II

    Olivier Benoist et Olivier Wittenberg. On the integral Hodge conjecture for real varieties, II. J. \'Ec. polytech. Math. , 7:373--429, 2020

    work page 2020

  8. [8]

    The C lemens- G riffiths method over non-closed fields

    Olivier Benoist et Olivier Wittenberg. The C lemens- G riffiths method over non-closed fields. Algebr. Geom. , 7(6):696--721, 2020

    work page 2020

Show all 40 references
  1. [9]

    Algebraic cycles and higher K -theory

    Spencer Bloch. Algebraic cycles and higher K -theory. Adv. in Math. , 61(3):267--304, 1986

    1986

  2. [10]

    Bochnak, M

    J. Bochnak, M. Coste et M.-F. Roy. G\'eom\'etrie alg\'ebrique r\'eelle , volume 12 de Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . Springer-Verlag, Berlin, 1987

    1987

  3. [11]

    Real algebraic geometry , volume 36 de Ergebnisse der Mathematik und ihrer Grenzgebiete (3)

    Jacek Bochnak, Michel Coste et Marie-Fran c oise Roy. Real algebraic geometry , volume 36 de Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . Springer-Verlag, Berlin, 1998

    1998

  4. [12]

    Rationality of singular cubic threefolds over R

    Ivan Cheltsov, Yuri Tschinkel et Zhijia Zhang. Rationality of singular cubic threefolds over R . arXiv:2411.14379 , 2024

    2024 arXiv

  5. [13]

    Birational invariants, purity and the G ersten conjecture

    Jean-Louis Colliot-Th\' e l\`ene. Birational invariants, purity and the G ersten conjecture. K -theory and algebraic geometry: connections with quadratic forms and division algebras ( S anta B arbara, CA , 1992) , volume 58 de Proc. Sympos. Pure Math. , pages 1--64. Amer. Math...

    1992

  6. [14]

    Parimala

    Jean-Louis Colliot-Th\' e l\`ene et R. Parimala. Real components of algebraic varieties and \'etale cohomology Invent. math. 101(1) : 81--99, 1990

    1990

  7. [15]

    Hypersurfaces quartiques de dimension 3: non-rationalit\' e stable

    Jean-Louis Colliot-Th\' e l\`ene et Alena Pirutka. Hypersurfaces quartiques de dimension 3: non-rationalit\' e stable. Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 49(2):371--397, 2016

    2016

  8. [16]

    Certaines fibrations en surfaces quadriques r\'eelles

    Jean-Louis Colliot-Th \'e l \`e ne et Alena Pirutka. Certaines fibrations en surfaces quadriques r\'eelles. arXiv:2406.00463 , 2024

    2024

  9. [17]

    Skorobogatov

    Jean-Louis Colliot-Th\'el\`ene et Alexei N. Skorobogatov. Groupe de C how des z\'ero-cycles sur les fibr\'es en quadriques. K -Theory , 7(5):477--500, 1993

    1993

  10. [18]

    Skorobogatov

    Jean-Louis Colliot-Th\' e l\`ene et Alexei N. Skorobogatov. The B rauer- G rothendieck group , volume 71 de Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics . Springer, Cham, 2021

    2021

  11. [19]

    Intersections of two quadrics and C h\^atelet surfaces

    Jean-Louis Colliot-Th\'el\`ene, Jean-Jacques Sansuc et Peter Swinnerton-Dyer. Intersections of two quadrics and C h\^atelet surfaces. I . J. reine angew. Math. , 373:37--107, 1987

    1987

  12. [20]

    Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale

    Annibale Comessatti. Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale. Math. Ann. , 73(1):1--72, 1912

    1912

  13. [21]

    Nash triviality in families of N ash manifolds

    Michel Coste et Masahiro Shiota. Nash triviality in families of N ash manifolds. Invent. math. , 108(2):349--368, 1992

    1992

  14. [22]

    Semialgebraic topology over a real closed field

    Hans Delfs et Manfred Knebusch. Semialgebraic topology over a real closed field. I. Paths and components in the set of rational points of an algebraic variety. Math. Z. , 177(1):107--129, 1981

    1981

  15. [23]

    Semialgebraic topology over a real closed field

    Hans Delfs et Manfred Knebusch. Semialgebraic topology over a real closed field. II . B asic theory of semialgebraic spaces. Math. Z. , 178(2):175--213, 1981

    1981

  16. [24]

    Rational equivalence on singular varieties

    William Fulton. Rational equivalence on singular varieties. Inst. Hautes \' E tudes Sci. Publ. Math. , (45):147--167, 1975

    1975

  17. [25]

    Intersection theory , volume 2 de Ergebnisse der Mathematik und ihrer Grenzgebiete

    William Fulton. Intersection theory , volume 2 de Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics . Springer-Verlag, Berlin, deuxi\`eme \'edition, 1998

    1998

  18. [26]

    Algebraic geometry

    Robin Hartshorne. Algebraic geometry . Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52

    1977

  19. [27]

    Rationality of even-dimensional intersections of two real quadrics

    Brendan Hassett, J\'anos Koll\'ar et Yuri Tschinkel. Rationality of even-dimensional intersections of two real quadrics. Comment. Math. Helv. , 97(1):183--207, 2022

    2022

  20. [28]

    Rationality of complete intersections of two quadrics over nonclosed fields

    Brendan Hassett et Yuri Tschinkel. Rationality of complete intersections of two quadrics over nonclosed fields. With an appendix by Jean-Louis Colliot-Th\'el\`ene. Enseign. Math. , 67(1-2):1--44, 2021

    2021

  21. [29]

    Formes quadratiques sur un corps , volume 15 de Cours Sp\'ecialis\'es

    Bruno Kahn. Formes quadratiques sur un corps , volume 15 de Cours Sp\'ecialis\'es . Soci\'et\'e Math\'ematique de France, Paris, 2008

    2008

  22. [30]

    Unramified cohomology of quadrics

    Bruno Kahn, Markus Rost et Ramdorai Sujatha. Unramified cohomology of quadrics. I . Amer. J. Math. , 120(4):841--891, 1998

    1998

  23. [31]

    Unramified cohomology of quadrics

    Bruno Kahn et Ramdorai Sujatha. Unramified cohomology of quadrics. II . Duke Math. J. , 106 :449--484, 2001

    2001

  24. [32]

    T. Y. Lam. Introduction to quadratic forms over fields . Graduate Studies in Mathematics, Vol. 67. American Mathematical Society, Providence, RI, 2005

    2005

  25. [33]

    A. S. Merkurjev. On the norm residue symbol of degree 2 . Dokl. Akad. Nauk SSSR , 261(3):542--547, 1981

    1981

  26. [34]

    Unramified elements in cycle modules

    Alexander Merkurjev. Unramified elements in cycle modules. J. Lond. Math. Soc. (2) , 78(1):51--64, 2008

    2008

  27. [35]

    Chow groups with coefficients

    Markus Rost. Chow groups with coefficients. Doc. Math. , 1:No. 16, 319--393, 1996

    1996

  28. [36]

    Real and \'etale cohomology , volume 1588 de Lecture Notes in Math

    Claus Scheiderer. Real and \'etale cohomology , volume 1588 de Lecture Notes in Math. . Springer-Verlag, Berlin, 1994. xxiv+273 pp

    1994

  29. [37]

    Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one

    Claus Scheiderer. Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one. Invent. math. , 125(2):307--365, 1996

    1996

  30. [38]

    A course in real algebraic geometry---positivity and sums of squares , volume 303 de Graduate Texts in Mathematics

    Claus Scheiderer. A course in real algebraic geometry---positivity and sums of squares , volume 303 de Graduate Texts in Mathematics . Springer, Cham, 2024

    2024

  31. [39]

    Unramified cohomology, algebraic cycles and rationality

    Stefan Schreieder. Unramified cohomology, algebraic cycles and rationality. Rationality of varieties , volume 342 de Progr. Math. , pages 345--388. Birkh\"auser/Springer, Cham, 2021

    2021

  32. [40]

    Theorie der quadratischen F ormen in beliebigen K \"orpern

    Ernst Witt. Theorie der quadratischen F ormen in beliebigen K \"orpern. J. reine angew. Math. , 176:31--44, 1937. otherlanguage

    1937

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.