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 $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ 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 $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ 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 ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.