GdT Géométrie Algébrique Complexe

Secant varieties of Grassmannians

by Vincenzo Galgano

Salle Cavaillès (IMT)

Salle Cavaillès



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 $\mathbb P(\bigwedge^kV)$. The action of $SL(V)$ on its irreducible representation $\bigwedge^{k}V$ 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 $\sigma(X)$.