On Kleiman–Piene's question for Gauss maps
On Kleiman–Piene's question for Gauss maps
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关于高斯图的 Kleiman-Piene 问题
DOI：
10.1112/s0010437x06002211
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发表时间：
2006
影响因子：
1.8
通讯作者：
Satoru Fukasawa
中科院分区：
文献类型：
--
作者：
Satoru Fukasawa
我们研究了一个费马超曲面$X_0^{p+1}+\dots+X_n^{p+1}=0 \subset \mathbf{P}^n$与$n \ge 3$和$\mathbf{P}^1$的乘积，通过Segre嵌入在$\mathbf{P}^{2n+1}$中，其中$p>0$是基场的特征。这种光滑变化是非自反的并且具有高斯映射，这是一种嵌入。这给出了一个否定的答案，下面的Kleiman-Piene问题在任何积极的特征：高斯映射的可分性是否意味着反思性？唯一已知的给出否定答案的平滑例子是Kaji在特征2中给出的。
We study the product of a Fermat hypersurface $X_0^{p+1}+\dots+X_n^{p+1}=0 \subset \mathbf{P}^n$ with $n \ge 3$ and $\mathbf{P}^1$, embedded in $\mathbf{P}^{2n+1}$ by Segre embedding where $p>0$ is the characteristic of the base field. This smooth variety is nonreflexive and has Gauss map which is an embedding. This gives a negative answer to the following Kleiman–Piene question in any positive characteristic: does the separability of the Gauss map imply reflexivity? The only known smooth examples, which give a negative answer, are given by Kaji in characteristic 2.