Rank 3 quadratic generators of Veronese embeddings

Let be a very ample line bundle on a projective scheme defined over an algebraically closed field with. We say that satisfies property if the homogeneous ideal of the linearly normal embedding can be generated by quadrics of rank less than or equal to. Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property. In this paper, we first prove that if then satisfies property for all and. We also investigate the asymptotic behavior of property for any projective scheme. Specifically, we prove that (i) if is-regular then satisfies property for all, and (ii) if is an ample line bundle on then satisfies property for all sufficiently large even numbers. These results provide affirmative evidence for the expectation that property holds for all sufficiently ample line bundles on, as in the cases of Green and Lazarsfeld's condition and the Eisenbud-Koh-Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513-539]. Finally, when we prove that fails to satisfy property for all.