## ON SECANT VARIETIES OF COMPACT HERMITIAN SYMMETRIC SPACES

Citations: | 5 - 2 self |

### BibTeX

@MISC{Landsberg_onsecant,

author = {J. M. Landsberg and Jerzy Weyman},

title = {ON SECANT VARIETIES OF COMPACT HERMITIAN SYMMETRIC SPACES},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three- with one exception, the secant variety of the 21-dimensional spinor variety in P 63, whose ideal is generated in degree four. We also discuss the coordinate ring of secant varieties of compact Hermitian symmetric spaces. 1.

### Citations

400 |
Groupes et Algèbres de Lie
- Bourbaki
- 1968
(Show Context)
Citation Context ...PA∗ × Q)) = σ(Seg(PA∗ × PW ∗ )). Notation. Let K, G, P, X be as in the first paragraph. We use German letters to denote Lie algebras associated to algebraic groups. We use the ordering of roots as in =-=[3]-=-. The fundamental weights and the simple roots of g are respectively denoted ωi and αi. Pk denotes the maximal parabolic of G obtained by deleting the root spaces corresponding to negative roots havin... |

163 |
algebra cohomology and the generalized Borel-Weil theorem
- Kostant, Lie
- 1961
(Show Context)
Citation Context ...G-module Vλ occurs in K[G] H with multiplicity equal to the number of H-fixed points in V ∗ λ . Proof. Since G/H ⊆ ˆσ(X) we obtain an inclusion K[ˆσ(X)] ⊆ K[G/H] by restricting functions on ˆσ(X). By =-=[9]-=-, K[G] = ⊕ {λ dominant}Vλ ⊗V ∗ λ and thus K[G]H = ⊕ {λ dominant}Vλ ⊗ (V ∗ λ )H . � Note that although K[G] H is equipped with a grading, we do not know of any way to recover the grading from this desc... |

117 |
Geometrische Methoden in der Invariantentheorie
- Kraft
- 1984
(Show Context)
Citation Context ...following standard fact: Proposition 8.2. Let G be an algebraic group and H a closed subgroup. Then we have the following equality of G-modules: K[G/H] = ⊕ + λ∈Λ Vλ ⊗V G ∗H λ . For a proof see, e.g., =-=[8]-=-, Theorem 3, Chapter II, section 3. Recall that if G/H ⊂ PV , then K[G/H] maps into K[G/H] by restriction of functions, and K[G/H] is equipped with a grading (that depends on the embedding). We do not... |

78 | The Penrose transform. Its interaction with representation theory - Baston, Eastwood - 1989 |

57 |
Tangents and secants of algebraic varieties
- Zak
- 1993
(Show Context)
Citation Context ...eight λ = [−1,0,0,1,0]. This calculation is similar to the above, but significantly easier because ξ is irreducible. 8. The coordinate ring of σ(X) The following proposition is essentially due to Zak =-=[21]-=-: Proposition 8.1. Let X = G/P ⊂ PV be a homogeneously embedded homogeneous variety. Let λ denote the highest weight and µ denote the lowest weight of V , and let vλ,vµ be corresponding weight vectors... |

55 | Algebraic geometry of Bayesian networks
- Garcia, Stillman, et al.
(Show Context)
Citation Context ...nd let σ(X) ⊂ PV denote its secant variety, the Zariski closure of the set of points on a secant line to X. Recently there has been interest in the ideals of secant varieties of homogeneous varieties =-=[11, 14, 15, 2, 5, 19, 7, 8]-=-, and this paper contributes to their study. If the ideal of a variety X is generated in degree two, the minimal possible degree of generators for the ideal of σ(X) is three, although in general one d... |

39 | On the ideals of secant varieties of Segre varieties
- Landsberg, Manivel
(Show Context)
Citation Context ...nd let σ(X) ⊂ PV denote its secant variety, the Zariski closure of the set of points on a secant line to X. Recently there has been interest in the ideals of secant varieties of homogeneous varieties =-=[11, 14, 15, 2, 5, 19, 7, 8]-=-, and this paper contributes to their study. If the ideal of a variety X is generated in degree two, the minimal possible degree of generators for the ideal of σ(X) is three, although in general one d... |

27 | The projective geometry of Freudenthal's magic square
- Landsberg, Manivel
(Show Context)
Citation Context ...W) Λ 3 RW S7 S6 Q 12 Spin(p ⊥ /ˆp) PA × Y P 1 × Q G(2,A) × YQ RA ⊗S Here, if Y = G/P ⊂ PW, YQ = G/P ′ is the variety obtained via Tits’ shadows that parametrizes a space of quadric sections of Y (see =-=[13]-=-). One takes the marked Dynkin diagram for Y ⊂ PW and looks for the largest subdiagram whose resulting marked diagram is a quadric hypersurface. The marked diagram of YQ is obtained by marking all nod... |

25 |
Cohomology of vector bundles and syzygies, Cambridge Tracts
- Weyman
- 2003
(Show Context)
Citation Context ...arieties. A key point is that when X ⊂ PV is homogeneous, σ(X) is the closure of the orbit of the sum of a highest weight vector and a lowest weight vector. We obtain our results using the methods of =-=[20]-=-, as described in Theorem 2.1 below, along with some new results about induced representations. In brief, in each case we obtain a desingularization of σ(X), by exploiting that fact that each X has a ... |

15 | Combinatorial secant varieties
- Sturmfels, Sullivant
(Show Context)
Citation Context ...nd let σ(X) ⊂ PV denote its secant variety, the Zariski closure of the set of points on a secant line to X. Recently there has been interest in the ideals of secant varieties of homogeneous varieties =-=[11, 14, 15, 2, 5, 19, 7, 8]-=-, and this paper contributes to their study. If the ideal of a variety X is generated in degree two, the minimal possible degree of generators for the ideal of σ(X) is three, although in general one d... |

14 | On the ideals and singularities of secant varieties of Segre varieties
- Landsberg, Weyman
(Show Context)
Citation Context |

14 | On the ideals of secant varieties to certain rational varieties
- Catalisano, Geramita, et al.
(Show Context)
Citation Context ...let σ(X) ⊂ PV denote its secant variety, the Zariski closure of the set of points on the secant lines to X. Recently there has been interest in the ideals of secant varieties of homogeneous varieties =-=[1, 3, 5, 6, 9, 12, 13, 17]-=-, and this paper contributes to their study. If the ideal of a variety X is generated in degree two, the minimal possible degree of generators for the ideal of σ(X) is three ([9] Cor. 3.2), although i... |

12 | Generalizations of Strassens equations for secant varieties of Segre varieties
- Landsberg, Manivel
(Show Context)
Citation Context |

12 |
a computer algebra package for Lie group computations,http://young.sp2mi.univ-poitiers.fr/ marc/LiE
- LiE
(Show Context)
Citation Context ... gr(ξ) as an f module and then compute the action of Zi0 to determine the coefficient on ωi0 for each irreducible f-module appearing. The f-module decomposition is straightforward with the aid of LiE =-=[16]-=-, keeping in mind that, if dim A = 2, then (2) (3) Λ k (A ⊗ (U ⊕ K)) = ⊕ a+b=kSa,bA ⊗ S 2 a ,1 b(U ⊕ K) = ⊕ a+b=kSa,bA ⊗ (S 2 a ,1 bU ⊕ S 2 a ,1 b−1U ⊕ S 2 a−1 ,1 b+1U ⊕ S 2 a−1 ,1 bU) One then uses L... |

9 | Invariants d’un sous–groupe unipotent maximal d’un groupe semi-simple, Annales de l’Institut Fourier - Brion - 1983 |

9 | Series of Lie groups
- Landsberg, Manivel
- 2004
(Show Context)
Citation Context ...y primitive module relative to S 3 W ′ ⊂ S 3 W. For S21W ′ ⊂ S21W, in each case there is a unique primitive module S1(Y ) which is S21W ∩ (I2(Y ) ⊗ W) ⊂ W ⊗3 because I2(Q) has no linear syzygies (see =-=[12]-=-, these are the modules called (UW)Aad in §2.5). The modules I2(σ(Y )) are all trivial modules except for V An ω6 for n ≥ 6 (and Λ 2 B ⊗ Λ 2 C for Y = Seg(PB ∗ × PC ∗ ). We now show there are no new g... |

8 | On the geometry of homogeneous varieties - Landsberg, Manivel |

7 |
The path model for representations of symmetrizable Kac-Moody algebras
- Littelmann
- 1994
(Show Context)
Citation Context ... W) ⊆ F(U) ⊗ F(W).s4 J.M. LANDSBERG, AND JERZY WEYMAN Proof. Since our modules are all eigenspaces for t c , it is sufficient to consider only the fdecomposition. The Littelmann path space (see e.g., =-=[17]-=-, §2) for a weight of f is contained in the path space for the same weight considered as a weight of g as the weight lattices are the same. Let λ,µ be the highest weights of U,W. The irreducible compo... |

5 |
Quievers and cohomology of homogenous vector bundles
- Ottaviani, Rubei
(Show Context)
Citation Context ... the ideal plus S21A ⊗ C, which is cancelled by its appearance in H1 , and it is the unique module appearing in H1 . By PropositionsON SECANT VARIETIES OF COMPACT HERMITIAN SYMMETRIC SPACES 11 6.7 in =-=[18]-=- cancellation occurs in the spectral sequence for the cohomology of Λ 3 ξ. The program we used for this calculation is publicly available at www.math.tamu.edu/∼robles. 7.3. Y = S5. We need to calculat... |

3 |
Id, Osculating varieties of Veronese varieties and their higher secant varieties
- Bernardi, Catalisano, et al.
(Show Context)
Citation Context |

2 |
Segre-Veronese embeddings of P 1 ×P 1 ×P 1 and their secant varieties
- Catalisano, Geramita, et al.
(Show Context)
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1 |
Scorza varieties and Jordan algebras, Indag
- Chaput
(Show Context)
Citation Context ...on lie in a unique J2(B) and the unique fiber is A ′ ⊗ J2(B). (The intersection Y ∩ PJ2(B) is the set of rank one elements in J2(B), i.e., a quadric of dimension dim B, see, e.g., [21], chapter VI or =-=[6]-=-.) For the case Y = S5, fix an isotropic line L ⊂ K 10 , the set {F ∈ S5 | L ⊂ F } is the shadow of L in S10 and the span of this shadow is the image of the fiber over [L] ∈ Q 8 in K 16 . Two general ... |

1 |
On spinor varieties, preprint
- Manivel
- 2008
(Show Context)
Citation Context ...answers a question posed in [9], Section 3. Recently, L. Manivel has made significant progress towards determining the generators of the ideals of secant varieties of spinor varieties in general, see =-=[15]-=-. While determining the generators of the ideals of secant varieties of higher rank CHSS seems out of reach at the moment, we show that for all CHSS other than spinor varieties, there are indeed gener... |