A LINEAR SYSTEM ON NARUKI'S MODULI SPACE OF MARKED CUBIC SURFACES

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E6.

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A FAMILY OF MARKED CUBIC SURFACES AND THE ROOT SYSTEM D5

International Journal of Mathematics ◽

10.1142/s0129167x07004163 ◽

2007 ◽

Vol 18 (05) ◽

pp. 505-525 ◽

Cited By ~ 1

Author(s):

ELISABETTA COLOMBO ◽

BERT VAN GEEMEN

Keyword(s):

Projective Space ◽

Root System ◽

Moduli Space ◽

Tangent Bundle ◽

Chow Group ◽

Cubic Surfaces ◽

Dimensional Projective Space

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

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The Action of the Weyl Group on the $$E_8$$ Root System

Graphs and Combinatorics ◽

10.1007/s00373-021-02315-8 ◽

2021 ◽

Author(s):

Rosa Winter ◽

Ronald van Luijk

Keyword(s):

Automorphism Group ◽

Weyl Group ◽

Root System ◽

Sufficient Conditions ◽

Maximal Clique ◽

Necessary And Sufficient Conditions ◽

Inner Product ◽

Colored Graphs ◽

Necessary And Sufficient ◽

Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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On Closed Subsets of Root Systems

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-050-4 ◽

1994 ◽

Vol 37 (3) ◽

pp. 338-345 ◽

Cited By ~ 6

Author(s):

D. Ž. Doković ◽

P. Check ◽

J.-Y. Hée

Keyword(s):

Weyl Group ◽

Root System ◽

Irreducible Component ◽

Product Space ◽

Root Systems ◽

Inner Product Space ◽

Inner Product ◽

Closed Sets ◽

Finite Dimensional ◽

Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

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Transformations depending on sets of associated points

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410002778x ◽

1952 ◽

Vol 48 (3) ◽

pp. 383-391

Author(s):

T. G. Room

Keyword(s):

Projective Space ◽

Projective Plane ◽

Three Dimensional ◽

Cubic Curves ◽

Birational Transformations ◽

Quadric Surfaces ◽

Base Points ◽

Dimensional Projective Space ◽

Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

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K 3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space

Inventiones mathematicae ◽

10.1007/s00222-003-0334-3 ◽

2004 ◽

Vol 156 (1) ◽

pp. 53-117 ◽

Cited By ~ 17

Author(s):

Ken-Ichi Yoshikawa

Keyword(s):

Moduli Space ◽

Automorphic Forms ◽

Analytic Torsion

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Wonderful compactifications of the moduli space of points in affine and projective space

European Journal of Mathematics ◽

10.1007/s40879-017-0170-4 ◽

2017 ◽

Vol 3 (3) ◽

pp. 520-564 ◽

Cited By ~ 1

Author(s):

Patricio Gallardo ◽

Evangelos Routis

Keyword(s):

Projective Space ◽

Moduli Space

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Codimension One Holomorphic Distributions on the Projective Three-space

International Mathematics Research Notices ◽

10.1093/imrn/rny251 ◽

2018 ◽

Vol 2020 (23) ◽

pp. 9011-9074 ◽

Cited By ~ 1

Author(s):

Omegar Calvo-Andrade ◽

Maurício Corrêa ◽

Marcos Jardim

Keyword(s):

Projective Space ◽

Moduli Space ◽

Tangent Bundle ◽

Projective Variety ◽

Arbitrary Degree ◽

Codimension One ◽

Quot Scheme ◽

Space Of Distributions ◽

Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

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The geometry and algebra of the representations of the Lorentz group

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1968.0056 ◽

1968 ◽

Vol 303 (1474) ◽

pp. 381-396 ◽

Cited By ~ 7

Keyword(s):

Projective Space ◽

Lorentz Group ◽

Point Of View ◽

Normal Curve ◽

Geometrical Representation ◽

Spinor Space ◽

Dimensional Projective Space ◽

And Algebra ◽

Rational Normal Curve ◽

Formal Point

In this paper a (2j + l)-spinor analysis is developed along the lines of the 2-spinor and 3-spinor ones. We define generalized connecting quantities A μv (j) which transform like (j, 0) ⊗ (j -1, 0) in spinor space and like second rank tensors under transformations in space-time. The general properties of the A uv are investigated together with algebraic relations involving the Lorentz group generators, J μv . The connexion with 3j symbols is discussed. From a purely formal point of view we introduce a geometrical representation of a (2j +1)-spinor as a point in a 2j dimensional projective space. Then, for example, the charge conjugate of a (2j + l)-spinor is just the polar of the corresponding point with respect to a certain rational, normal curve in the projective space. It is suggested that this representation will prove useful.

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