On the rank index of some quadratic varieties

Regarding the generating structure of the homogeneous ideal of a projective variety $X\subset {\mathbb{P}}^r$, we define the rank index of $X$ to be the smallest integer $k$ such that $I(X)$ can be generated by quadratic polynomials of rank at most $k$. Recently it is shown that every Veronese embedding has rank index $3$ if the base field has characteristic $\mathrm{\ne }2,3$. In this paper, we introduce some basic ways of how to calculate the rank index and find its values when $X$ is some other classical projective varieties such as rational normal scrolls, del Pezzo varieties, Segre varieties and the Plücker embedding of the Grassmannian of lines.