We discuss various results on Hilbert schemes of lines and conics andautomorphism groups of smooth Fano threefolds with Picard rank 1. Besides ageneral review of facts well known to experts, the paper contains some newresults, for instance, we give a description of the Hilbert scheme of conics onany smooth Fano threefold of index 1 and genus 10. We also show that the actionof the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1)on an irreducible component of its Hilbert scheme of lines (respectively,conics) is faithful if the anticanonical class of $X$ is very ample with apossible exception of several explicit cases. We use these faithfulness resultsto prove finiteness of the automorphism groups of most Fano threefolds andclassify explicitly all Fano threefolds with infinite automorphism group. Wealso discuss a derived category point of view on the Hilbert schemes of linesand conics, and use this approach to identify some of them.
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机译：我们讨论有关皮卡德等级为1的光滑Fano三重线的Hilbert方案的曲线和圆锥形以及自同构群的各种结果。除了对专家众所周知的事实进行一般性回顾之外，本文还包含一些新结果，例如，我们给出了Hilbert方案的描述。锥在索引1和属10的任何光滑Fano三倍上都是圆锥的。我们还证明了Fano在索引的Hilbert方案的不可约分量上分别是索引2的三倍$ X $（分别为圆锥）的自同构群的作用。 ）如果$ X $的反规范类非常充实，则可能是忠实的，但有几个明确的情况可能除外。我们使用这些忠实性结果来证明大多数Fano三重折叠同构群的有限性，并用无限的同构群将所有Fano三重折叠显式分类。我们还将讨论关于线和圆锥线的希尔伯特方案的派生类别观点，并使用这种方法来识别其中的一些。
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