On the Rank Index of Some Quadratic Varieties

A projective variety X⊂ P^r is said to satisfy property QR(k) if its homogeneous ideal can be generated by quadratic polynomials of rank at most k. We define the rank index of X to be the smallest integer k such that X satisfies property QR(k). Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property QR(4). Recently, it is shown in [19] that every Veronese embedding has rank index 3 if the base field has characteristic ≠ 2 , 3 . In this paper, we find the rank index of X when it is some other classical projective variety such as rational normal scrolls, Segre varieties, Plücker embeddings of the Grassmannians of lines and del Pezzo varieties.