Toroidal embedding of Chevalley groups over $\mathbb{Z}$

Shang Li

REVIEW 3 major objections 4 minor 11 references

Toroidal embedding of Chevalley groups over $\mathbb{Z}$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any fan supported in the negative Weyl chamber, this paper constructs a scheme X_Σ over Spec(Z) with a G×_Z G action whose base change to every algebraically closed field is the classical equivariant toroidal embedding of G_k.

desk verdict A serious, likely correct construction of universal toroidal embeddings over Z for arbitrary fans, with the main caveat being a black-boxed representability theorem from a companion paper. read the letter →

arxiv 2506.02638 v1 pith:SWWYYB4Y submitted 2025-06-03 math.AG

Shang Li This is my paper

classification math.AG MSC 14L1514M2514M2714L3020G35

keywords toroidalembeddingsChevalleygroupschemessplitreductiverationalactionsfansinthenegativeWeylchambertoricvarietiesintegralmodelsalgebraicspaces

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

This paper establishes an integral, or 'universal', version of equivariant toroidal embeddings for split reductive group schemes over the integers. For every fan in the negative Weyl chamber, it constructs a scheme X_Σ over Spec(Z) equipped with an action of G×_Z G that extends left and right translation of G, and whose base change to any algebraically closed field is the classical toroidal embedding of G_k determined by the same fan. Because the classical classification is combinatorial and characteristic-independent, such integral models are expected to specialize fiberwise; the paper proves they exist. The construction works uniformly for arbitrary fans, including non-affine and non-projective ones, and gives quasi-projectivity for single cones plus combinatorial smoothness and properness criteria.

What carries the argument

The load-bearing object is the fppf quotient sheaf X_σ = (G ×_S Ω_σ ×_S G)/∼_{A_σ}, built from the rational action A_σ: G ×_S Ω_σ ×_S G ⇢ Ω_σ that extends the two-sided translation of G on its open cell Ω_G ≅ U^- × T × U^+. The rational action is reconstructed step by step from explicit root-subgroup formulas, using the morphisms f_i and f to swap positive and negative root subgroups through T_σ, and its definition domain is forced to contain e × Ω_σ × e. Quoted theorems on rational actions then promote the quotient sheaf to an algebraic space with a G×G action, and the sheaf-theoretic description is what makes the special-fiber statement transparent: base change commutes with the quotient, so (X_σ)_k is literally the quotient that the classical embedding satisfies. Gluing via open immersions X_τ ↪ X_σ for faces τ ⊂ σ assembles X_Σ.

What would settle it

Compute the definition domain of A_σ for a non-smooth cone σ over Spec(Z), for instance the cone generated by (1,1) in a rank-two cocharacter lattice; if e × Ω_σ × e is not contained in it, or if the equivalence relation on G × Ω_σ × G is not an fppf equivalence relation with representable diagonal, then X_σ is not an algebraic space and Theorem 1.1 fails.

Watch

Extended reading notes

Core claim

The central discovery is that the classical toroidal embedding of a reductive group can be lifted to a single flat scheme over Spec(Z) that remembers the whole family of special fibers. For a single cone σ, the paper constructs X_σ as the fppf quotient sheaf (G ×_S Ω_σ ×_S G)/∼_{A_σ}, where Ω_σ = U^- ×_S T_σ ×_S U^+ is the big-cell toric scheme and A_σ is the rational action of G×G on Ω_σ extending the group law on the open cell G. Quoting rational-action theorems, the quotient is shown to be a quasi-projective algebraic space over S and, étale locally on the base, a scheme; gluing these along face inclusions yields X_Σ. The geometric fiber over an algebraically closed field k is identified with the classical toroidal embedding G_{k,σ} by matching the big-cell open subschemes and the G×G orbits. The paper also proves that X_Σ is smooth exactly when every cone is generated by a subset of a basis of the cocharacter lattice, and proper exactly when the Weyl-group saturation of the fan is complete.

Load-bearing premise

The construction depends on the quoted rational-action theorems applying to the toric big-cell scheme Ω_σ, which is not a group scheme, and that application is not verified in this paper.

Editorial extensions

If this is right

For every fan Σ in the negative Weyl chamber there is now a single flat-scheme model over Spec(Z) whose geometric fibers are the classical toroidal embeddings, so the combinatorial classification holds uniformly across all characteristics.
Each single-cone embedding X_σ is quasi-projective over the base, so the universal models are scheme-theoretically reasonable, not merely algebraic spaces.
Smoothness and properness of X_Σ can be read off the fan: smooth iff every cone is generated by a subset of a basis of X_*(T), proper iff W·Σ is complete.
The construction is uniform for arbitrary fans and does not require the embedding to be affine or projective, unlike previous integral models built from canonical bases.
The quotient-sheaf description supplies a functoriality X_{σ1} → X_σ for cone inclusions, so the whole fan is glued from compatible local models.

Reading between the lines

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

If the quoted rational-action theorems hold in the generality assumed, the same quotient-sheaf recipe should produce integral models for other equivariant embeddings with a big-cell structure, such as spherical varieties or symmetric-space compactifications.
The sheaf-theoretic construction may still work even when T_σ is not smooth over Z, since it never uses a group scheme structure on Ω_σ; concrete rank-two examples with non-smooth cones would test this directly.
One can compare these Z-models with the canonical-basis models from a recent preprint for affine and projective embeddings; where both exist, they should agree on the big cell, which would give an independent check of the fibral identification.
A direct calculation of the definition domain of A_σ in the SL_2 case would give an explicit local picture of X_σ and clarify how the rational-action axioms from the companion paper are used.

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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

Dear [Colleague],

You should know this about arXiv:2506.02638: it proves a genuinely new existence theorem—universal equivariant toroidal embeddings over Spec Z for split reductive group schemes, for arbitrary fans supported in the negative Weyl chamber. The statement is new: prior work covered only the wonderful compactification ([Li23]) and affine/projective cases ([BS25]), while the classical BK05 classification was over algebraically closed fields. The construction is uniform and sheaf-theoretic: for each cone, X_σ is an fppf quotient sheaf (G×Ω_σ×G)/∼, with Ω_σ the toric big cell, and the paper shows this is a quasi-projective scheme over Z and glues to X_Σ. Theorem 4.4, where the rational action is built step-by-step with explicit formulas, is the technical heart, and it looks right.

The main soft spot is structural: the step that turns the quotient sheaf into an algebraic space is imported wholesale from the author's companion paper [Li25] (Theorems 3.6 and 3.7 here). The present text does not verify that the hypotheses of those theorems hold for Ω_σ, which is flat and finitely presented but emphatically not a group scheme. The stated theorems are general enough to cover it, but a referee cannot check this without reading [Li25] side-by-side. This is addressable—the author should either state the hypotheses and check them, or include the necessary arguments. It is not a fatal gap, but it is a real dependency.

The stress-test's specific worry—that only the identity-section domain condition is shown—is not quite right: Theorem 4.4 constructs the rational action on a dense open and verifies it agrees with the group law, which gives associativity. The real issue is simply that the representability result is black-boxed.

Two smaller things. Lemma 4.8, the fibral identification with the classical embedding, is compressed: the monomorphism to G_{k,σ} is clear from the rational action, but the surjectivity argument needs a sentence or two. Proposition 5.6's smoothness/properness criteria are stated with only brief sketches; if they're meant as results, they deserve fuller proofs.

If [Li25]'s results are as general as claimed, the paper is correct. The proof structure is sound, the citations to SGA3/BLR/EGA are appropriate, and I see no circularity or overreach. I'd send this to a serious referee: the theorem is important enough to merit the time, and the known gaps are all fillable. It's also a good reading-group paper for people working on integral compactifications.

Referee Report

3 major / 4 minor

Summary. The paper develops a theory of toroidal embeddings for Chevalley group schemes over Z, in analogy with the classical classification of equivariant toroidal embeddings of reductive groups over algebraically closed fields. For a split reductive group G over a scheme S, a maximal split torus T contained in a Borel B, and a fan Σ in the negative Weyl chamber, the main theorem (Theorem 5.4) asserts the existence of a scheme X_Σ over Z with a G×G action extending the left and right translations of G, whose base change to any algebraically closed field is the classical toroidal embedding associated with Σ. The construction proceeds by first defining a rational action A_σ of G×G on the big cell Ω_σ = U^- × T_σ × U^+ (Theorem 4.4), then forming an fppf quotient sheaf X_σ = (G×Ω_σ×G)/∼ (Construction 4.6), which is claimed to be an algebraic space by results quoted from the companion paper [Li25]. The single-cone case is then glued along face inclusions to obtain X_Σ. The paper also states quasi-projectivity of the single-cone pieces (Theorem 4.9) and combinatorial criteria for smoothness and properness (Proposition 5.6).

Significance. If the gaps identified below are resolved, this paper would provide a uniform integral model for all toroidal embeddings of reductive groups over Z, going beyond earlier constructions for affine and projective cases and for the wonderful compactification. The explicit formulas for the rational action on the big cell (Lemma 4.1, Lemma 4.3, Theorem 4.4) are a concrete and useful contribution, and the sheaf-theoretic approach is natural. The paper is also careful to check its construction against the classical classification over algebraically closed fields (Lemma 4.8, Theorem 5.4), which is the right benchmark. However, the central representability step is imported from an unpublished companion paper without verification of its hypotheses, and the properness argument in Proposition 5.6 is not convincing. These issues are load-bearing, so the main theorem is not established within the present text.

major comments (3)

[§3, Construction 4.6] The assertion that X_σ is an algebraic space rests entirely on Theorem 3.6 ([Li25, Cor. 5.9, Prop. 5.11]) and Theorem 3.7 ([Li25, Thm. 5.6]), neither of which is proved in this manuscript. For the specific scheme Y = Ω_σ, the paper verifies only the condition e×Ω_σ×e ⊂ Dom(A_σ) (Theorem 4.4). It does not verify the hypotheses needed to apply [Li25, Thm. 5.6], for instance that the two projections from the graph Γ of φ(g_1,g_2,ω) = A_σ(g_1^{-1},A_σ(g_2,ω)) to G×Ω_σ form an fppf equivalence relation, nor that the quotient by this graph relation is isomorphic to the quotient sheaf defined by ∼_A. Since the algebraic-space structure is the foundation for Lemma 4.8, Theorem 4.9, and the gluing in Section 5, the central theorem is unsupported unless the results of [Li25] are supplied or their hypotheses are explicitly checked for this Y.
[§5, Proposition 5.6(2)] The proof of the properness criterion is not valid as written. It states that the criterion holds over an algebraically closed field by [BK05, Prop. 6.2.3(iv)] and then concludes that X_Σ is proper over Z because, by Lemma 4.8, all geometric fibers are proper and geometrically connected, citing [EGA IV3, Cor. 15.7.11]. Properness is not a fibral property: a separated finite-type morphism with proper geometric fibers need not be proper, and the cited corollary does not provide such a criterion. A direct argument, e.g., via the valuative criterion, is required. This gap affects the 'if' direction of the claimed combinatorial characterization of properness.
[§4.3, Theorem 4.9] The proof that X_σ is quasi-projective is too terse and depends on unstated assumptions. It shows that π: X_σ → X is affine by reducing, after étale descent to a strictly Henselian base, to showing that χ: Ω_σ → π^{-1}(Ω) is an isomorphism, using the fibral criterion [EGA IV4, Cor. 17.9.5]. The hypotheses of that fibral criterion (such as finite presentation and properness of the relevant morphism) are not checked, and the equality of the open immersion with the classical big-cell identification is asserted via Lemma 4.8 and [BK05, Prop. 6.2.3(i)] without spelling out the compatibility. This needs a more detailed proof, especially because the quasi-projectivity of X_σ is what turns the algebraic space of Construction 4.6 into a scheme.

minor comments (4)

[§4.2, Lemma 4.2] The notation f_1 in the proof of Lemma 4.2 is confusing: the text writes "we define f_1 in a similar way" but the displayed definition then uses f_j for the induced automorphism of V_j. Please use distinct symbols for the composed rational map and the individual simple-reflection maps.
[§4.3, Lemma 4.8] In the proof of Lemma 4.8, the map ξ sending (g_1,ω,g_2) to g_1·ω·g_2 is called a monomorphism, but the well-definedness of ξ with respect to the equivalence relation ∼_A is not explicitly checked. It follows from the compatibility of A_σ with the classical action on each geometric fiber, but this should be stated.
[§5, Remark 5.5] The phrase "the only way I can imagine" is informal for a journal article; the remark also asserts without proof that the direct fppf quotient sheaf for a general fan Σ is an algebraic space, which again depends on the unverified results from [Li25].
[Throughout] The arXiv text contains numerous transcription artifacts (e.g., '−/∫hortrightarrow', 'Ş', 'Gk−/∫hortrightarrowpXΣqk'), which should be corrected in the final version. Some displayed equations in the proof of Lemma 4.1 have potential typographical issues in the coordinates; please proofread the formulas against a standard Chevalley-system reference.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular step: the quotient-sheaf construction is independent; the only caveat is a load-bearing but non-circular self-citation to [Li25] for representability.

full rationale

The derivation chain is not circular. Theorem 1.1 is obtained by gluing the fppf quotient sheaves X_σ = (G ×_S Ω_σ ×_S G)/∼_{A_σ} of Construction 4.6, where the rational action A_σ is built in Theorem 4.4 from the Chevalley system and toric data, not from the existence of toroidal embeddings over Z. The classical classification [BK05] is used only after the construction, as an external benchmark to identify the geometric fibers (Lemma 4.8, Theorem 5.4), and no fitted parameter is relabeled as a prediction. The one genuine caveat is that the representability of X_σ as an algebraic space is imported from the author's companion paper [Li25] via Theorems 3.6 and 3.7; the present text does not prove those theorems or verify all their hypotheses for Y = Ω_σ. This is a load-bearing self-citation and a verification gap, but it is not a circularity: [Li25] is a separately argued rational-action theorem whose assumptions do not include the target universal-embedding statement, and nothing in the quoted reduction assumes Theorem 1.1. Hence the paper's central claim retains independent content, and the circularity score is low.

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

The construction has no free parameters and invents no new entities: it is a pure existence and construction theorem with no fitting. It rests on (i) standard fppf and algebraic-space machinery, (ii) SGA3 structure theory for split reductive groups over Z, (iii) the classical BK05 classification as an external benchmark, (iv) the author's [Li25] rational-action theorems used as a black box (the main non-standard debt: they must apply to the toric scheme Ω_σ, which is not a group scheme), and (v) standard EGA and BLR fibral criteria cited without verification. The heaviest burden is (iv): it is recent work by the same author, and its hypotheses are not re-checked here.

assumptions (5)

standard math Fppf quotient sheaf formalism and Artin's criterion: the quotient of an algebraic space by an fppf equivalence relation is an algebraic space ([LM00, corollaire (10.4)], [SP, 04S6 (2)], [Ana73, théorème 3.1.1]).
Used to prove Theorem 3.6 (algebraicity of the quotient sheaf) in Section 3; standard in the Laumon-Moret-Bailly and Stacks Project literature.
standard math SGA3 structure theory of split reductive group schemes over Z: Existence and Isomorphism Theorems, Chevalley systems, root subgroups, and the big cell open immersion ([SGA 3III, exposés XXII-XXV]).
Invoked throughout Section 4.1 (open immersion Ω_G, root data, simple reflections n_{α_i}) and in Lemma 4.1 (conjugation by n_{α_i}).
standard math Classical bijection between fans supported in the negative Weyl chamber and equivariant toroidal embeddings over an algebraically closed field ([BK05, Proposition 6.2.3, 6.2.4], Theorem 2.2 of this paper).
This is the external benchmark: Lemma 4.8 and Theorem 5.4 identify the fibers (X_Σ)_k with the classical objects, inheriting the classical uniqueness.
domain assumption The rational-action theorems of the author's separate preprint [Li25] (its Theorem 5.6, Corollary 5.9, Proposition 5.11, Lemmas 5.2 and 5.3) are correct and apply to the non-group scheme Y = Ω_σ = U^- ×_S T_σ ×_S U^+ with the rational action A_σ.
Section 3 states these results without proof, and Construction 4.6 applies them to Ω_σ; the algebraic-space and scheme representability arguments (Theorem 4.9, Lemma 4.7) depend on them.
domain assumption Standard fibral criteria from EGA and BLR: [EGA IV4, corollaire 17.9.5] (fibral criterion for isomorphisms and open immersions), [EGA IV3, corollaire 15.7.11] (properness from proper connected fibers), [BLR90, section 2.3 Proposition 5 and section 2.5 Proposition 6] (existence of sections and flat…
Used in Lemmas 4.2(2), 4.3, 4.7, 4.8, 5.2 and Proposition 5.6; the hypotheses of each cited statement are not verified in the text.

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Pith. "Pith review of Toroidal embedding of Chevalley groups over $\mathbb{Z}$." pith.science (2026). https://pith.science/paper/SWWYYB4Y

@misc{pith2026250602638,
  author       = {Pith},
  title        = {Pith review of: Toroidal embedding of Chevalley groups over $\mathbbZ$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWWYYB4Y}},
  note         = {Machine review of arXiv:2506.02638}
}

abstract

The classification of equivariant toroidal embeddings of a reductive group over an algebraically closed field is combinatorial and does not depend on the characteristic of the base field. This suggests that there should exist ``universal'' toroidal embeddings for a Chevalley group scheme over $\mathbb{Z}$ which specialize to classical toroidal embeddings via base change. In this paper, we establish the existence of ``universal'' equivariant toroidal embeddings for split reductive group schemes over $\mathbb{Z}$. We also discuss several geometric properties of these embeddings.

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Works this paper leans on

11 extracted references · 10 canonical work pages

[3]

Study of a Z-form of the coordinate ring of a reductive group

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Se- ries. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2000, pp. xii+208. isbn: 3-540-65761-4. [Lus09] G. Lusztig. “Study of a Z-form of the coordinate ring of a reductive group.” In: J. Amer. Math. Soc.22.3 (2009), pp. 739–769.issn: 089...

work page doi:10.1090/s0894-0347-08- 2009
[6]

Reductive group schemes

Soc. Math. France, Paris, 1995, Exp. No. 219, 219–234.isbn: 2-85629-039-6. [Con14] Brian Conrad. “Reductive group schemes.” In: Autour des schémas en groupes. Vol. I. Vol. 42/43. Panor. Synthèses. Soc. Math. France, Paris, 2014, pp. 93–444. [DP83] C. De Concini and C. Procesi. “Complete symmetric varieties.” In: Invariant theory (Montecatini, 1982). Vol

work page 1995
[8]

Séminaire de Géométrie Algébrique du Bois Marie 1962–64

Documents Mathématiques (Paris) [Mathematical Documents (Paris)]. Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 19...

work page 1962
[15]

An introduction to the theory of toric varieties, Translated from the Japanese

Ergebnisse der Mathe- matik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. An introduction to the theory of toric varieties, Translated from the Japanese. Springer-Verlag, Berlin, 1988, pp. viii+212.isbn: 3-540-17600-4. [SGA 3II] Schémas en groupes. II: Groupes de type multiplicatif, et structure des sché- mas en groupes généra...

work page 1988
[101]

Supplément au Bull. Soc. Math. France. Soc. Math. France, Paris, 1973, pp. 5–79. doi: 10.24033/msmf.109 . url: https://doi.org/10.24033/ msmf.109. [Bible] Séminaire C. Chevalley, 1956–1958. Classification des groupes de Lie algébriques. 2 vols. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958, ii+166 + ii+122 pp. (mimeographed). [BK05] Michel Br...

work page doi:10.24033/msmf.109 1973
[152]

Demazure et A

Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Dirigé par M. Demazure et A. Grothendieck. Springer-Verlag, Berlin-New York, pp. ix+654. [SGA 3III] Philippe Gille and Patrick Polo, eds. Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs. Vol

work page 1962
[339]

An equivariant compactification for adjoint reductive group schemes

Springer-Verlag, Berlin-New York, 1973, pp. viii+209. [Li23] Shang Li. “An equivariant compactification for adjoint reductive group schemes.” In: arXiv preprint arXiv:2308.01715(2023). [Li25] Shang Li. Wonderful embedding for group schemes in the Bruhat–Tits theory

work page Pith review arXiv 2023
[996]

Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes

Lecture Notes in Math. Springer, Berlin, 1983, pp. 1–44. isbn: 3-540-12319-9. doi: 10 . 1007 / BFb0063234. url: https : //doi.org/10.1007/BFb0063234. [EGA II] A. Grothendieck and J. Dieudonné. “Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes.” In:Inst. Hautes Études Sci. Publ. Math.8 (1961), p. 222.issn: 0...

work page doi:10.1007/bfb0063234 1961

Show all 11 references

[1064]

Hermann & Cie, Paris, 1948, p

[Current Scientific and Industrial Topics]. Hermann & Cie, Paris, 1948, p

1948
[2024]

Variétés abéliennes et courbes algébriques

[Wei48] André Weil. Variétés abéliennes et courbes algébriques. Vol. 8 (1946). Publications de l’Institut de Mathématiques de l’Université de Strasbourg [Publications of the Mathematical Institute of the University of Strasbourg]. Actualités Scientifiques et Industrielles, No

1946
[2025]

Certains schémas de groupes semi-simples

arXiv: 2505.01173 [math.RT] . url: https://arxiv.org/ abs/2505.01173. [Che95] Claude Chevalley. “Certains schémas de groupes semi-simples.” In: Séminaire Bourbaki, Vol

arXiv

Pith tools

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

[1] [3]

Study of a Z-form of the coordinate ring of a reductive group

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Se- ries. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2000, pp. xii+208. isbn: 3-540-65761-4. [Lus09] G. Lusztig. “Study of a Z-form of the coordinate ring of a reductive group.” In: J. Amer. Math. Soc.22.3 (2009), pp. 739–769.issn: 089...

work page doi:10.1090/s0894-0347-08- 2009

[2] [6]

Reductive group schemes

Soc. Math. France, Paris, 1995, Exp. No. 219, 219–234.isbn: 2-85629-039-6. [Con14] Brian Conrad. “Reductive group schemes.” In: Autour des schémas en groupes. Vol. I. Vol. 42/43. Panor. Synthèses. Soc. Math. France, Paris, 2014, pp. 93–444. [DP83] C. De Concini and C. Procesi. “Complete symmetric varieties.” In: Invariant theory (Montecatini, 1982). Vol

work page 1995

[3] [8]

Séminaire de Géométrie Algébrique du Bois Marie 1962–64

Documents Mathématiques (Paris) [Mathematical Documents (Paris)]. Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 19...

work page 1962

[4] [15]

An introduction to the theory of toric varieties, Translated from the Japanese

Ergebnisse der Mathe- matik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. An introduction to the theory of toric varieties, Translated from the Japanese. Springer-Verlag, Berlin, 1988, pp. viii+212.isbn: 3-540-17600-4. [SGA 3II] Schémas en groupes. II: Groupes de type multiplicatif, et structure des sché- mas en groupes généra...

work page 1988

[5] [101]

Supplément au Bull. Soc. Math. France. Soc. Math. France, Paris, 1973, pp. 5–79. doi: 10.24033/msmf.109 . url: https://doi.org/10.24033/ msmf.109. [Bible] Séminaire C. Chevalley, 1956–1958. Classification des groupes de Lie algébriques. 2 vols. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958, ii+166 + ii+122 pp. (mimeographed). [BK05] Michel Br...

work page doi:10.24033/msmf.109 1973

[6] [152]

Demazure et A

Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Dirigé par M. Demazure et A. Grothendieck. Springer-Verlag, Berlin-New York, pp. ix+654. [SGA 3III] Philippe Gille and Patrick Polo, eds. Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs. Vol

work page 1962

[7] [339]

An equivariant compactification for adjoint reductive group schemes

Springer-Verlag, Berlin-New York, 1973, pp. viii+209. [Li23] Shang Li. “An equivariant compactification for adjoint reductive group schemes.” In: arXiv preprint arXiv:2308.01715(2023). [Li25] Shang Li. Wonderful embedding for group schemes in the Bruhat–Tits theory

work page Pith review arXiv 2023

[8] [996]

Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes

Lecture Notes in Math. Springer, Berlin, 1983, pp. 1–44. isbn: 3-540-12319-9. doi: 10 . 1007 / BFb0063234. url: https : //doi.org/10.1007/BFb0063234. [EGA II] A. Grothendieck and J. Dieudonné. “Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes.” In:Inst. Hautes Études Sci. Publ. Math.8 (1961), p. 222.issn: 0...

work page doi:10.1007/bfb0063234 1961