Secants of spinor varieties

par Vincenzo Galgano

mercredi 8 mars 2023, 15:45 → 16:45 Europe/Paris

Salle Cavaillès (IMT)

The spin group $Spin(V)$ is the double universal cover of the group $SO(V)$. The spin representations are the only fundamental representations of $Spin(V)$ which do not come from representations of $SO(V)$. The spinor variety $\mathbb{S}$ is the closed orbit of the highest weight vector in the projectivized spin representation $\mathbb{P}(\mathrm{\Delta })$. The action of $Spin(V)$ on $\mathrm{\Delta }$ induces an action on the secant variety of lines $\sigma (\mathbb{S})\subset \mathbb{P}(\mathrm{\Delta })$. Similarly to the Grassmannian case, we determine the orbits in $\sigma (\mathbb{S})$ together with their dimensions and the inclusions among their closures. We apply nonabelian apolarity for determining which points lie on a unique secant or tangent line and we compute the singular locus of $\sigma (\mathbb{S})$ via the notion of secant bundle.