On automorphisms of Enriques surfaces
On automorphisms of Enriques surfaces
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DOI：
10.1007/bf01388499
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发表时间：
1984-02
影响因子：
3.1
通讯作者：
I. Dolgachev;I. Dolgachev
中科院分区：
文献类型：
--
作者：
I. Dolgachev;I. Dolgachev
特征为4=2的代数闭域k上的Enrique曲面是非奇异射影曲面F，满足Hi(F，(Gv)=H2(F，or)=0，2Kv=0.由扭类Kv定义的F的未分枝二重覆盖是K3曲面F，即HI(F，(GR)=0，Kr=0)的非奇异射影曲面。Enriques曲面的研究等价于具有不动点自由对合z的K3曲面的研究.特别地，F的自同构群Aut(F)与群Aut(ff，z)/(Z)同构，其中Aut(F，z)是z在Ft的自同构群Aut(F)中的中心化子.在复数域k=~的情况下，Aut(F)的研究是基于I.Piatetski-Shapiro和I.Shafarevich在[19]中证明的K3曲面的整体Torelli定理。由此定理得出：群Aut(Ff)同构于商群O(Pic(F))/W，其中O(Pic(F))是f的Picard格的正交群，W是反射到非奇异有理曲线族中所生成的正规子群.对于“一般的”Enrique曲面F，这个定理允许计算Aut(F)(见[3]和[17]，其中这个结果没有明确说明)。对于任意的F，F和ff之间的关系无济于事，因为很难计算z在Pie(IF)中的作用。然而，通过其他方法，我们可以证明与Piatetski-Shapiro和Shafarevich的结果类似的结果：定理。设F是Enriques曲面，HR=Pic(F)/Tors。设W~是反射到F上的非奇异有理曲线族的正交群O(Hf)的子群，G是WF~生成的子群，Aut(F)在O(HR)中的像Aut(F)*.则W~是G的正规子群，G是W~与Aut(F)*的半直积。此外，如果k=Lu，则G在O(Hf)中具有有限指数。
An Enriques surface over an algebraically closed field k of characteristic 4= 2 is a nonsingular projective surface F with Hi (F,(gv)= H2 (F, Or)= 0, 2Kv= 0. The unramified double cover of F defined by the torsion class K v is a K3-surface F, a nonsingular projective surface with HI (F,(gr)= 0, Kr= 0. The study of Enriques surfaces is equivalent to the study of K3-surfaces with a fixed-point-free involution z. In particular, the automor_phism group Aut (F) of F is isomorphic to the group Aut (ff, z)/(z), where Aut (F, z) is the centralizer of z in the automorphism group Aut (F) of ft. In the case k=~, the field of complex numbers, the study of Aut (F) is based on the Global Torelli Theorem for K3-surfaces proven by I. Piatetski-Shapiro and I. Shafarevich in [19]. It follows from this theorem that up to a finite group the group Aut (ff) is isomorphic to the quotient group O (Pic (F))/W, where O (Pic (F)) is the orthogonal group of the Picard lattice of ff and W its normal subgroup generated by the reflections into the classes of nonsingular rational curves. For a" generic" Enriques surface F this theorem allows to compute Aut (F)(see [3] and also [17], where this result is not stated explicitly). For an arbitrary F the relation between F and ff does not help, since it is very difficult to compute the action of z in Pie (if). However, by other means, we can prove the following analog of Piatetski-Shapiro and Shafarevich's result:Theorem. Let F be an Enriques surface and Hr= Pic (F)/Tors. Let W~ be the subgroup of the orthogonal group O (HF) generated by reflections into the classes of nonsingular rational curves on F and G be the subgroup generated by Wf~ and the image Aut (F)* of Aut (F) in O (Hr). Then W~ is a normal subgroup of G and G is the semi-direct product of W~ and Aut (F)*. Moreover, if k= lU then G is of finite index in O (HF).