On linear sections of the spinor tenfold II

Laurent Manivel (IMT); Yingqi Liu (AMSS)

REVIEW 3 major objections 5 minor 1 cited by

On linear sections of the spinor tenfold II

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Codimension-four linear sections of the spinor tenfold have the same GIT moduli space as Kummer surfaces.

desk verdict A genuinely new bridge between spinor tenfold sections and Kummer surfaces, but the core Cartan-subspace verification is delegated to an unavailable preprint, so the theorem is conditional on an external black box. read the letter →

arxiv 2504.21056 v1 pith:3K3BBWTY submitted 2025-04-29 math.AG

Yingqi Liu (AMSS) , Laurent Manivel (IMT) This is my paper · ORCID

classification math.AG MSC 14J4514J2814L2417B20

keywords spinortenfoldlinearsectionsKummersurfacesGITmodulispaceVinbergtheorycomplexreflectiongroupCartansubspaceFanovarieties

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 that the GIT moduli space of codimension-four linear sections of the spinor tenfold — the ten-dimensional Fano variety in $P^{15}$ whose smaller slices realize many prime Fano threefolds, polarized K3 surfaces, and genus-seven curves — is the weighted projective space wP(2,3,5,6), and that this same moduli space is the moduli space of Kummer surfaces. The identification runs through a Z/4-grading of the exceptional Lie algebra e8 whose degree-one piece is $C^{4}$ ⊗ Δ−, the vector space whose projectivized rank-four elements parametrize the sections. A four-dimensional Cartan subspace c carries a complex reflection group Wc of order 46080, and the quotient P(c)/Wc is the desired weighted projective space. The paper also constructs explicit models of the two isolated special sections, one with automorphism group GL2 and one with SL2×SL2; the latter compactifies SL2×SL2/μ10, in analogy with the Mukai–Umemura threefold. If correct, this gives a Lie-theoretic explanation of Klein's classical construction of Kummer surfaces from quadratic line complexes.

What carries the argument

The central object is the Cartan subspace c ⊂ $C^{4}$⊗Δ− arising from the Z/4-grading e8 = g0⊕g1⊕g2⊕g3 with g0 = sl4×so10 and g1 = $C^{4}$⊗Δ−. It is a four-dimensional abelian subspace on which the little Weyl group Wc acts as the complex reflection group G31; the quotient P(c)/Wc is wP(2,3,5,6). The machinery has two engines: Vinberg's Jordan-class theory, which reduces the orbit classification in the non-finite orbit space to semisimple elements in c plus finitely many nilpotent orbits, and the classical Kummer geometry, encoded in the spinor quadratic complex and the maps Θ and K. The reflection hyperplanes of Wc form the Klein configuration of 60 planes, and the Wc-representations U5, U9, V5 (spanned by products of 4, 8, and 12 hyperplane equations) realize the Specht modules that produce the Igusa quartic and the Segre cubic.

What would settle it

Compute the Hilbert series of the invariant ring of Spin10 on the affine cone over G(4,Δ−) and compare it with the invariant ring of Wc on c; any discrepancy in the degrees 8, 12, 20, 24 would disprove the isomorphism. Alternatively, find a semistable rank-four subspace K ⊂ Δ− whose orbit closure does not meet the image of the Cartan subspace c.

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Extended reading notes

Core claim

The paper's central discovery is a precise isomorphism of GIT moduli spaces: G(4,Δ−)//Spin10 ≃ P(c)/Wc ≃ wP(2,3,5,6), where c is a Cartan subspace of the Z/4-graded Lie algebra e8 with g1 ≅ $C^{4}$ ⊗ Δ−, and Wc ≅ G31 is the complex reflection group of order 46080. This quotient is also the moduli space of Kummer surfaces. The proof combines the semisimple/nilpotent classification of Vinberg θ-representations (via de Graaf's classification of the 145 nilpotent orbits) with explicit geometric maps: a degree-16 morphism Θ from P(c) to the Igusa quartic CR4, whose fibers over a general point are the 16 singular points of a Kummer surface, and a degree-256 rational map K from P(c) to the Segre cubic S3 sending a section to the Kummer surface it produces. The paper further proves that the unique section with SL2×SL2 action is a compactification of SL2×SL2/μ10.

Load-bearing premise

The load-bearing premise is that every GIT-equivalence class of codimension-four sections is represented by a semisimple element in the Cartan subspace; the paper proves the vector-space version but only asserts the Grassmannian version.

Editorial extensions

If this is right

Semisimple codimension-four sections are classified by the nine flat types of the Klein arrangement, with singular sections lying on 30 lines in P(c).
The general codimension-four section has automorphism group F_2^4, correcting an earlier claim of F_2^2.
Adding a general nilpotent part can smooth a section whose semisimple part is singular; only three nilpotent orbits give smooth sections.
The codimension-four sections admit a universal family over the Igusa quartic via Coble quadrics, whose singular loci are the associated abelian surfaces.
The quantum cohomology of a smooth codimension-four section is presented by two relations and is generically semisimple, matching the Gamma conjecture eigenvalue prediction.

Reading between the lines

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

If the moduli isomorphism holds, then GIT stability for codimension-four sections should be readable entirely from the semisimple Jordan type; one testable consequence is that the K-polystable compactifications of the moduli space compare with the quotient compactifications wP(2,3,5,6) and S3/S6 along the lines the paper suggests.
The SL2×SL2 section may serve as a six-dimensional analogue of the Mukai–Umemura threefold; its explicit GKM graph and quantum relations could support a conjecture about eigenvalues of quantum multiplication for higher-dimensional Fano manifolds of high index.
The combinatorial identification of the spinorial 16_6 configuration with the Kummer configuration suggests that other minuscule representations could produce new configurations of points and tropes, with corresponding moduli isomorphisms for linear sections.

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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

Colleague,

The main theorem here—that the GIT moduli of codimension four spinor tenfold sections equals the GIT moduli of Kummer surfaces, both wP(2,3,5,6)—is new and, if right, a real bridge between Vinberg theory and classical Kummer geometry. The architecture is convincing, but the load-bearing stone is not visible. Proposition 5.5, the assertion that the explicit 4-plane c is a Cartan subspace, is delegated to [dG25], an unpublished preprint. The step from the GIT quotient of C4⊗Δ− by SL4×Spin10 to the Grassmannian quotient G(4,Δ−)//Spin10 is also asserted rather than proved. Both matter: without the first, P(c)/Wc need not be the quotient one thinks; without the second, the claimed moduli identification doesn’t follow from the Vinberg statement.

That said, the paper earns a lot of credit. The block/pentad combinatorics, the maps to the Igusa quartic and Segre cubic, the explicit SL2×SL2 section and its compactification of SL2×SL2/μ10, and the honest correction of [DM22] are all coherent and genuinely informative. The appendix gives substantial data for independent checking, and the authors don’t hide the use of external computations.

My advice: send it out, but ask for a rewrite that either proves Proposition 5.5 directly (commutation of the p_i, semisimplicity, generic stabilizer dimension zero) or makes the de Graaf preprint available and states precisely which statements rely on it. The Grassmannian comparison can be closed with a standard frame-bundle paragraph. The computational shortcuts are acceptable in this area; the problem is that the central identification rests on an unavailable source.

Serious referee: yes. The paper is important enough and mostly coherent. I’d read a revised version carefully.

Referee Report

3 major / 5 minor

Summary. The paper studies linear sections of the spinor tenfold X ⊂ P(Δ+), with the main focus on codimension-four sections. It uses the Z4-grading of e8 with g1 ≃ C4 ⊗ Δ+ and Vinberg's theory of graded Lie algebras. An explicit Cartan subspace c and the associated little Weyl group Wc = G31 are introduced, and the paper claims that the GIT moduli space of codimension-four sections is G(4,Δ−)//Spin10 ≃ P(c)/Wc ≃ wP(2,3,5,6), which is also the moduli space of Kummer surfaces. The argument passes through the spinor quadratic complex, producing a degree-16 morphism Θ: P(c) → CR4 (the Castelnuovo-Richmond/Igusa quartic) and a degree-256 map K: P(c) ⇢ S3 (the Segre cubic), thereby recovering classical Kummer combinatorics via Hudson forms, blocks, pentads, and synthemes. The paper also classifies orbits in C4 ⊗ Δ− (including 145 nilpotent orbits), describes the discriminant locus as 30 lines, gives explicit models for special sections with automorphism groups GL2 and SL2 × SL2, identifies the latter with a compactification of SL2 × SL2/μ10, and computes ordinary and quantum cohomology by GKM localization.

Significance. If the main theorem holds, this is a substantial and attractive result: it establishes a precise bridge between linear sections of the spinor tenfold and classical Kummer surface geometry, and it gives an explicit weighted-projective-space model for the moduli space. The explicit formulas for Θ, for the Kummer equations, for the blocks/pentads, and for the GKM graph are valuable and could be independently verified. However, the proof rests at several load-bearing points on external computational and unpublished input, especially de Graaf's preprint [dG25], as well as on Macaulay, GAP, and Magma computations. These dependencies are not inherently disqualifying, but the manuscript currently leaves the reader unable to verify the central claims without trusting those sources. The gaps appear fixable, so the result is promising but the proof needs to be made self-contained or explicitly conditional.

major comments (3)

[§5.3, Proposition 5.5] The assertion that the four-dimensional space c spanned by p1,...,p4 in display (8) is a Cartan subspace is delegated to [dG25]. No verification is given that [pi,pj] = 0, that the elements of c are semisimple, or that c is maximal, for instance by checking that a generic element of c has zero-dimensional stabilizer in SL4 × Spin10. This is load-bearing because Vinberg's isomorphism C4 ⊗ Δ+//(SL4 × Spin10) ≃ c/Wc is valid only when c is a Cartan subspace; if c were merely an abelian subspace of semisimple elements, the quotient c/Wc would not automatically be the GIT quotient of the representation. Please provide a complete proof of Proposition 5.5, or a detailed computational verification included in the paper or an appendix.
[Introduction (Theorem) and §5.1] The theorem identifies G(4,Δ−)//Spin10 with P(c)/Wc, but the paper only states the affine quotient C4 ⊗ Δ+//(SL4 × Spin10) ≃ c/Wc. The invariant-theoretic comparison between the Grassmannian quotient and the projective tensor quotient is not written out: the semistability of full-rank tensors for the SL4 × Spin10 action and the matching of polarizations between Sym^d(∧4 Δ−) and the SL4-invariant part of Sym^{4d}((C4 ⊗ Δ−)∨) need to be justified. In addition, the sections are parametrized by Δ− while the Cartan subspace c is constructed in C4 ⊗ Δ+; the dual Cartan subspace c′ is introduced only informally in the remark after Proposition 5.5, and Proposition 5.14 does not by itself establish the GIT identification for G(4,Δ−). Please supply a detailed proof of the Grassmannian/tensor comparison and of the Δ−/c′ version needed for the theorem.
[§6, Proposition 6.7 and Table (14); §5.3] The nilpotent orbit classification (145 orbits), the stabilizer types in Table (13), and the list of 60 reflection hyperplanes are all sourced to [dG25], while several other central computational assertions are justified only by 'Macaulay' or 'GAP', for example the dimension-and-degree statement in the proof of Proposition 6.1 and the index computation in the proof of Proposition 5.22. Since the classification of smooth sections, the description of the discriminant locus, and the identification of the special automorphism groups depend on these data, the computations need to be reproduced in the manuscript or made available in a verifiable form, such as scripts or data files. Alternatively, the main theorem should be explicitly stated as conditional on the results of [dG25].

minor comments (5)

[§5.3 and Appendix C] The list of 60 hyperplanes appears to contain a typo: item (12) repeats the equation a3 − a4 already given as item (8); it should presumably read a3 − i a4, consistent with item (11).
[§5.3, Corollary 5.13] The notation Pc is used for P(c); please unify the notation throughout the paper.
[§4.2] The polynomial c(κ) introduced in the discussion of ap(AP(κ)) = c(κ)κ conflicts with the later use of c for the Cartan subspace; a different letter would avoid ambiguity.
[Throughout] There are several typographical slips: 'Sheppard-Todd' should be 'Shephard-Todd', 'spnots' should be 'spinors', 'Bilanicki-Birula' and 'Byalinicki-Birula' should be 'Bialynicki-Birula'.
[Introduction] The main theorem is stated in the Introduction as an unnumbered display; numbering it would make the later references to it clearer.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the central GIT-moduli isomorphism rests on Vinberg theory, classical Kummer geometry, and de Graaf's independent classification, with only minor non-load-bearing self-citations.

full rationale

The derivation chain is not circular. The main identification C4⊗Δ+//(SL4×Spin10) ≃ c/Wc is quoted from Vinberg's classical theory of graded Lie algebras [Vin76], and the explicit four-dimensional subspace c is asserted to be a Cartan subspace in Proposition 5.5 with the proof delegated to W. de Graaf's independent classification [dG25]; this is an external, verifiability-relevant input, not a self-citation and not fitted to the paper's target result. The subsequent identifications P(c)/Wc ≃ wP(2,3,5,6) and the isomorphism with the Kummer moduli space are obtained through the Castelnuovo-Richmond quartic and Segre cubic, using classical references such as [Dol12], [GD94], and [ST54], not through the authors' own prior results. The paper's self-citations, including [DM22], [Man19], [BBFM24], and the in-preparation [BBFM25], are used for background, for routine invariant-theoretic facts, or to correct an earlier claim about generic automorphism groups; none of them is load-bearing for the main GIT quotient isomorphism. In particular, the correction of the generic automorphism group from F2^2 to F2^4 does not enter the proof that P(c)/Wc is the moduli space of codimension four sections. No equation or construction in the paper is equivalent, by definition, to its own input, and no fitted parameter is renamed as a prediction. The only notable gap is the delegation of the Cartan-subspace verification and the orbit classification to the external preprint [dG25]; that is a reproducibility and verification concern, not a circularity concern, and accordingly it does not raise the circularity score beyond a minor self-citation level.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central argument relies on standard results in Vinberg theory, Shephard-Todd theory, and classical Kummer geometry, plus the external classification [dG25]. No free parameters are fitted to data; the constants in Section 6.4.2 are arbitrary and do not affect isomorphism types. No new entities are postulated.

assumptions (4)

standard math Vinberg theory of Z-graded Lie algebras provides the Cartan subspace c and the little Weyl group Wc for the Z4-grading of e8 with g1 = C4 ⊗ Δ−.
Invoked in Section 5.1 and Appendix B; this is established theory [Vin76], and the specific case is read from Vinberg's table (Case 11).
standard math Shephard-Todd classification: the reflection group Wc is G31 of order 46080 with invariant degrees 8, 12, 20, 24.
Used in Section 5.1 and Corollary 5.13 to identify the quotient with wP(2,3,5,6).
domain assumption Classical classification and geometry of 16_6 configurations and Kummer surfaces, as presented by Gonzalez-Dorrego, Hudson, and Dolgachev, are correct.
Used in Sections 4.4, 5.2, and 5.6 to identify configurations and the moduli spaces CR4 and S3.
domain assumption de Graaf's classification [dG25] of the 145 nilpotent orbits and Jordan classes in C4⊗Δ− is complete and correct.
The entire classification of smooth sections in Section 6.2 and the automorphism group table in Section 6.3 rely on this external preprint, which is not reproduced.

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Pith. "Pith review of On linear sections of the spinor tenfold II." pith.science (2026). https://pith.science/paper/3K3BBWTY

@misc{pith2026250421056,
  author       = {Pith},
  title        = {Pith review of: On linear sections of the spinor tenfold II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3K3BBWTY}},
  note         = {Machine review of arXiv:2504.21056}
}

abstract

Following previous work by A. Kuznetsov, we study the Fano manifolds obtained as linear sections of the spinor tenfold in $\mathbb{P}^{15}$. Up to codimension three there are finitely many such sections, up to projective equivalence. In codimension four there are three moduli, and this family is particularly interesting because of its relationship with Kummer surfaces on the one hand, and a grading of the exceptional Lie algebra $\mathfrak{e}_8$ on the other hand. We show how the two approaches are intertwined, and we prove that codimension four sections of the spinor tenfolds and Kummer surfaces have the very same GIT moduli space. The Lie theoretic viewpoint provides a wealth of additional information. In particular we locate and study the unique section admitting an action of $SL_2\times SL_2$; similarly to the Mukai-Umemura variety in the family of prime Fano threefold of genus 12, it is a compactification of a quotient by a finite group.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Algebraic cycles and Fano threefolds of genus 7
math.AG 2026-08 conditional novelty 7.0 of 10

A very general prime Fano threefold of genus 7 admits no multiplicative Chow-Künneth decomposition, because the explicit cycle Z_Y on Y×Y is Abel-Jacobi trivial but non-zero in the Chow group.

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Works this paper leans on

18 extracted references · 17 canonical work pages · cited by 1 Pith paper

[1]

Benedetti, M

[BBFM24] V. Benedetti, M. Bolognesi, D. Faenzi, and L. Maniv el, The Coble quadric , Forum Math. Sigma 12 (2024), Paper No. e63,

work page 2024
[2]

Ciubotaru, Spin representations of Weyl groups and the Springer corres pondence, J

[Ciu12] D. Ciubotaru, Spin representations of Weyl groups and the Springer corres pondence, J. Reine Angew. Math. 671 (2012), 199–222. [CS19] I. Cheltsov and C. Shramov, Finite collineation groups and birational rigidity , Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 71,

work page 2012
[5]

Dedieu and L

[DM22] Th. Dedieu and L. Manivel, On the automorphisms of Mukai varieties , Math. Z. 300 (2022), no. 4, 3577–3621. [Dol04] I.V. Dolgachev, Abstract conﬁgurations in algebraic geometry , The Fano Conference, Univ. Torino, 2004, pp. 423–462. [Dol12] , Classical algebraic geometry. A modern view , Cambridge,

work page 2022
[6]

conﬁgurations and geometry of Kummer surfaces in P3, Mem. Amer. Math. Soc. 107 (1994), no. 512, vi+101. [GGI16] S. Galkin, V. Golyshev, and H. Iritani, Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures, Duke Math. J. 165 (2016), no. 11, 2005–2077. [GH94] Ph. Griﬃths and J. Harris, Principles of algebraic geometry , Wiley Classics L...

work page 1994
[9]

Howard, J

[HMSV08] B. Howard, J. Millson, A. Snowden, and R. Vakil, A description of the outer automorphism of S6, and the invariants of six points in projective space , J. Combin. Theory Ser. A 115 (2008), no. 7, 1296–1303. [HNS00] K. Hulek, I. Nieto, and G. K. Sankaran, Heisenberg-invariant Kummer surfaces , Proc. Edinburgh Math. Soc. (2) 43 (2000), no. 2, 425–43...

work page 2008
[13]

[KW13] , Geometry of orbit closures for the representations associa ted to gradings of lie algebras of types e7, arXiv e-print math.AG/1301.0720,

arXiv
[14]

Morris and P

[MM07] A.O. Morris and P. Mwamba, Macdonald representations of complex reﬂection groups , S´ em. Lothar. Combin. 52 (2004/07), Art. B52g,

work page 2004
[17]

Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commuta- tive algebra, Vol

[Muk88] S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commuta- tive algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 357–377. [Muk92] , Fano 3-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989 ), London Math. Soc. Lecture Note Ser., vol. 179, Cambridge, 1992, pp. 255–2

work page 1988

Show all 18 references

[63]

Math., vol

[Muk12] , Igusa quartic and Steiner surfaces , Compact moduli spaces and vector bundles, Contemp. Math., vol. 564, Amer. Math. Soc., 2012, pp. 205–210. [New81] P. E. Newstead, Invariants of pencils of binary cubics , Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 201–209...

1981
[389]

Furushima and N

[FN89] M. Furushima and N. Nakayama, The family of lines on the Fano threefold V5, Nagoya Math. J. 116 (1989), 111–122. [GD94] M.R. Gonzalez-Dorrego, (16 ,

1989
[1956]

Chevalley, The algebraic theory of spinors and Cliﬀord algebras , Springer, 1997, Collected works

[Che97] C. Chevalley, The algebraic theory of spinors and Cliﬀord algebras , Springer, 1997, Collected works. Vol

1997
[1972]

Sato and T

[SK77] M. Sato and T. Kimura, A classiﬁcation of irreducible prehomogeneous vector spac es and their relative invariants , Nagoya Math. J. 65 (1977), 1–155. [ST54] G.C. Shephard and J.A. Todd, Finite unitary reﬂection groups , Canad. J. Math. 6 (1954), 274–304. [Vin76] E.B. Vi...

1977
[1994]

Goresky, R

[GKM98] M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. [GSW13] Laurent Gruson, Steven V. Sam, and Jerzy Weyman, Moduli of abelian varieties, Vinberg θ-groups, and free re...

1998
[1996]

Kimura and I

[KO82] T. Kimura and I. Ozeki, On the microlocal structure of a regular prehomogeneous vec tor space associated with Spin(10) × GL(3), Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 6, 239–242. [Kon16] S. Kond¯ o, The Igusa quartic and Borcherds products , K3 surfaces and ...

1982
[2011]

[KW12] , Geometry of orbit closures for the representations associa ted to gradings of lie algebras of types e6, f4 and g2, arXiv e-print math.AG/1201.1102,

arXiv
[2012]

Fu and J.-M

[FH20] B. Fu and J.-M. Hwang, Euler-symmetric projective varieties , Algebr. Geom. 7 (2020), no. 3, 377–

2020
[2013]

Landsberg and L

[LM03] J.M. Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieti es, Comment. Math. Helv. 78 (2003), no. 1, 65–100. 48 Y. LIU AND L. MANIVEL [Mac72] I.G. Macdonald, Some irreducible representations of Weyl groups, Bull. London Math. Soc. 4 (1972...

2003
[2025]

[BFM20] Ch. Bai, B. Fu, and L. Manivel, On Fano complete intersections in rational homogeneous var ieties, Math. Z. 295 (2020), no. 1-2, 289–308. ON LINEAR SECTIONS OF THE SPINOR TENFOLD II 47 [Cat23] F. Catanese, Kummer quartic surfaces, strict self-duality, and more , The ar...

2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Benedetti, M

[BBFM24] V. Benedetti, M. Bolognesi, D. Faenzi, and L. Maniv el, The Coble quadric , Forum Math. Sigma 12 (2024), Paper No. e63,

work page 2024

[2] [2]

Ciubotaru, Spin representations of Weyl groups and the Springer corres pondence, J

[Ciu12] D. Ciubotaru, Spin representations of Weyl groups and the Springer corres pondence, J. Reine Angew. Math. 671 (2012), 199–222. [CS19] I. Cheltsov and C. Shramov, Finite collineation groups and birational rigidity , Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 71,

work page 2012

[3] [5]

Dedieu and L

[DM22] Th. Dedieu and L. Manivel, On the automorphisms of Mukai varieties , Math. Z. 300 (2022), no. 4, 3577–3621. [Dol04] I.V. Dolgachev, Abstract conﬁgurations in algebraic geometry , The Fano Conference, Univ. Torino, 2004, pp. 423–462. [Dol12] , Classical algebraic geometry. A modern view , Cambridge,

work page 2022

[4] [6]

conﬁgurations and geometry of Kummer surfaces in P3, Mem. Amer. Math. Soc. 107 (1994), no. 512, vi+101. [GGI16] S. Galkin, V. Golyshev, and H. Iritani, Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures, Duke Math. J. 165 (2016), no. 11, 2005–2077. [GH94] Ph. Griﬃths and J. Harris, Principles of algebraic geometry , Wiley Classics L...

work page 1994

[5] [9]

Howard, J

[HMSV08] B. Howard, J. Millson, A. Snowden, and R. Vakil, A description of the outer automorphism of S6, and the invariants of six points in projective space , J. Combin. Theory Ser. A 115 (2008), no. 7, 1296–1303. [HNS00] K. Hulek, I. Nieto, and G. K. Sankaran, Heisenberg-invariant Kummer surfaces , Proc. Edinburgh Math. Soc. (2) 43 (2000), no. 2, 425–43...

work page 2008

[6] [13]

[KW13] , Geometry of orbit closures for the representations associa ted to gradings of lie algebras of types e7, arXiv e-print math.AG/1301.0720,

arXiv

[7] [14]

Morris and P

[MM07] A.O. Morris and P. Mwamba, Macdonald representations of complex reﬂection groups , S´ em. Lothar. Combin. 52 (2004/07), Art. B52g,

work page 2004

[8] [17]

Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commuta- tive algebra, Vol

[Muk88] S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commuta- tive algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 357–377. [Muk92] , Fano 3-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989 ), London Math. Soc. Lecture Note Ser., vol. 179, Cambridge, 1992, pp. 255–2

work page 1988