Numerically trivial automorphisms of Enriques surfaces in characteristic 2
Numerically trivial automorphisms of Enriques surfaces in characteristic 2
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DOI：
10.2969/jmsj/78867886
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发表时间：
2017-09
影响因子：
0.7
通讯作者：
I. Dolgachev;G. Martin
中科院分区：
文献类型：
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作者：
I. Dolgachev;G. Martin
一个代数曲面$S的自同构称为上同调(数值上)平凡的，如果它作用在第二个$L的上同调群(这个群的模扭子群)上。将S.Mukai和Y.Nankawa的结果推广到任意特征{p>；0}，证明了上同调平凡自同构群$\rm{aut}{\rm{ct}}(S)$是$\leq 2$阶的，如果$S$不是超奇异的.如果$p=2$且$S$是超奇异的，我们证明了$rm{aut}{\rm{ct}}(S)$是1，2，3，5，7，11$中的奇数阶$n循环群或$8$的四元数群，并对所有的例外情况作了明确的刻画。如果$K_S\neq 0$，我们还证明了数值平凡自同构群$\rm{aut}_{\rm{nt}}(S)$是阶为$\leq 4$的循环群的一个子群，除非$p=2$，其中$\rm{aut}_{\rm{nt}}(S)$是秩为$\leq 2$的$2$-初等群的子群。
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order $8$ and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\rm{Aut}_{\rm{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\leq 4$ unless $p = 2$, where $\rm{Aut}_{\rm{nt}}(S)$ is a subgroup of a $2$-elementary group of rank $\leq 2$.