Rank 3 quadratic generators of Veronese embeddings

Abstract

Let L be a very ample line bundle on a projective scheme X defined over an algebraically closed field k with char k not equal 2. We say that (X, L) satisfies property QR(k) if the homogeneous ideal of the linearly normal embedding X subset of PH0(X, L) can be generated by quadrics of rank less than or equal to k. Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property QR(4). In this paper, we first prove that if char k not equal 3 then (P-n, O-Pn(d)) satisfies property QR(3) for all n >= 1 and d >= 2. We also investigate the asymptotic behavior of property QR(3) for any projective scheme. Specifically, we prove that (i) if X subset of PH0(X, L) is m-regular then (X, L-d) satisfies property QR(3) for all d >= m, and (ii) if A is an ample line bundle on X then (X, A(d)) satisfies property QR(3) for all sufficiently large even numbers d. These results provide affirmative evidence for the expectation that property QR(3) holds for all sufficiently ample line bundles on X, as in the cases of Green and Lazarsfeld's condition N-p 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 char k = 3 we prove that (P-n, O-Pn(2)) fails to satisfy property QR(3) for all n >= 3.