SIGMA 5 (2009), 078, 22 pages      arXiv:0904.0565      https://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 D_n, 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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