On the Rank Index of Some Quadratic Varieties

초록

A projective variety X subset of 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 not equal 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, Plucker embeddings of the Grassmannians of lines and del Pezzo varieties.

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