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 $\mathbb{k}$ with $\mathrm{char}\mathbb{k}\ne 2$. 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 $\le k$. 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 $\mathrm{char}\mathbb{k}\ne 3$ then $({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(d))$ satisfies property $\mathsf{QR}(3)$ for all $n\ge 1$ and $d\ge 2$. We also investigate an asymptotic behavior of property $\mathsf{QR}(3)$ for any projective scheme. Namely, 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 $d\ge m$ 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 number $d$. These results provide an 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-Lazarsfeld's condition ${\mathrm{N}}_p$ and Eisenbud-Koh-Stillman's determininantal presentation in [EKS88]. Finally, when $\mathrm{char}\mathbb{k}=3$ we prove that $({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(2))$ fails to satisfy property $\mathsf{QR}(3)$ for all $n\ge 3$.

24 pages

Cited by in corpus (1)

• On the rank index of projective curves of almost minimal degree