关于可分高次Gauss映射
On separable higher Gauss maps
摘要：
本文研究了F.~意义下的$m$阶高斯映射L.~代数闭域上的射影簇$X \subset \mathbb{P}^N$的Zak。对所有具有$n:=\dim(X) \leq m < N$的整数$m$，证明了当$m$阶Gauss映射可分时，一般切$m$ -平面在$X$上的接触轨迹是线性簇.我们还证明了，对于光滑的$X$与$n < N-2$，$(n+1)$ -th高斯映射是双有理的，如果它是可分的，除非$X$是Segre嵌入$\mathbb{P}^1 \times \mathbb{P}^n \subset \mathbb{P}^{2n-1}$。这与L有关。Ein对特征为零的小对偶品种的分类。
英文摘要：
We study the $m$-th Gauss map in the sense of F.~L.~Zak of a projective variety $X \subset \mathbb{P}^N$ over an algebraically closed field in any characteristic. For all integer $m$ with $n:=\dim(X) \leq m < N$, we show that the contact locus on $X$ of a general tangent $m$-plane is a linear variety if the $m$-th Gauss map is separable. We also show that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss map is birational if it is separable, unless $X$ is the Segre embedding $\mathbb{P}^1 \times \mathbb{P}^n \subset \mathbb{P}^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.
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