Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 078, 22 pages      arXiv:0904.0565      http://dx.doi.org/10.3842/SIGMA.2009.078
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

On Spinor Varieties and Their Secants

Laurent Manivel
Institut Fourier, Université de Grenoble I et CNRS, BP 74, 38402 Saint-Martin d'Hères, France

Received April 03, 2009, in final form July 21, 2009; Published online July 24, 2009

Abstract
We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type Dn, cubic equations exist if and only if n ≥ 9. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.

Key words: spinor variety; spin representation; secant variety; Freudenthal variety.

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