论文标题
Veronese嵌入的3级二次发电机
Rank 3 Quadratic Generators of Veronese Embeddings
论文作者
论文摘要
让$ l $是一个非常宽敞的线条捆绑包，用于射影方案$ x $在代数封闭的字段$ \ bbbk $上定义的，$ {\ rm char}〜\ bbbk \ bbbk \ neq 2 $。我们说，如果线性正常嵌入$ x \ subset \ mathbb {p} h^0（x，x，l）$的均质理想的均相理想，则$（x，l）$满足属性$ \ mathsf {qr}（k）$。许多古典品种，例如segre-veronese嵌入，高度的高度滚动和曲线满足属性$ \ mathsf {qr}（4）$。 在本文中，我们首先证明，如果$ {\ rm char}〜\ bbbk \ neq 3 $，则$（\ mathbb {p}^n，\ mathcal {o} _ {\ mathbb {p}^n}^n}^n}^n}（d））$ 2 $。我们还研究了任何投影方案的属性$ \ mathsf {qr}（3）$的渐近行为。也就是说，我们证明$（i）$如果$ x \ subset \ mathbb {p} h^0（x，x，l）$是$ m $ - then $（x，l^d）$满足属性$ \ mathsf {qr}（qr}（qr}（qr}（qr}（qr}（qr）属性$ \ mathsf {qr}（3）$，用于所有足够大的数字$ d $。这些结果为期望属性$ \ mathsf {qr}（3）$提供了肯定的证据，例如$ x $上的所有充分的线条包，例如Green-Lazarsfeld的条件$ \ Mathrm {n} _p $ and Eisenbud-koh-koh-stillman的确定性介绍。 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 $n \geq 3$.
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 $\leq k$. 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 $n \geq 1$ and $d \geq 2$. 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 $d \geq m$ 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 ${\rm char}~\Bbbk = 3$ we prove that $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (2))$ fails to satisfy property $\mathsf{QR}(3)$ for all $n \geq 3$.
