Secant varieties of Grassmannians

par Vincenzo Galgano

lundi 20 févr. 2023, 14:30 → 15:30 Europe/Paris

Salle Cavaillès (IMT)

The secant variety of lines $\sigma (X)$ of a projective variety $X\subset {\mathbb{P}}^m$ is the union of secant and tangent lines to $X$ in ${\mathbb{P}}^m$. We consider $X=Gr(k,V)$ the Grassmannian of $k$-planes in a complex vector space $V$, embedded via Plucker in $\mathbb{P}(\stackrel{k}{\bigwedge }V)$. The action of $SL(V)$ on its irreducible representation $\stackrel{k}{\bigwedge }V$ induces an action on $\sigma (X)$. In this talk we analyze the $SL(V)$-orbits in $\sigma (X)$, determining their representatives, the inclusions and the dimensions of their closures. Moreover, via the technique of  nonabelian apolarity, we determine which points of $\sigma (X)$ lie on a unique bisecant or tangent line to $X$. Finally, we use the notion of secant bundle to determine the singular locus of $\sigma (X)$.