We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of 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 except for some explicit cases.We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use it to identify some of them.

中文翻译：

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Fano的线，圆锥和自同构群的Hilbert方案三重

我们讨论有关Picard秩1的光滑Fano三重性的线和圆锥形的Hilbert方案和自同构群的各种结果。除了对专家熟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，我们对关于指数1和属10的任何光滑Fano三倍的圆锥形的Hilbert锥方案。我们还表明，指数2的Fano三倍X的自同构群（分别为1）对其线的Hilbert方案的不可约成分的作用（如果X的反正则类是忠实的除了一些明确的情况外，它非常充裕。我们使用这些忠实性结果来证明大多数Fano三倍自同构群的有限性，并使用无限自同构群将所有Fano三倍显式明确地分类。我们还将讨论关于直线和圆锥线的希尔伯特方案的派生类别观点，并使用它来识别其中的一些观点。