Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

TL;DR

This paper studies the Hilbert schemes of lines and conics and the automorphism groups of smooth Fano threefolds with Picard rank 1, providing new descriptions, faithfulness results, and classifications of automorphism groups.

Contribution

It offers new descriptions of Hilbert schemes of conics on Fano threefolds of index 1 and genus 10, and classifies Fano threefolds with infinite automorphism groups.

Findings

Faithfulness of automorphism group actions on Hilbert schemes in most cases

Finiteness of automorphism groups for most Fano threefolds

Explicit classification of Fano threefolds with infinite automorphism groups

Abstract

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with 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 with a possible exception of several 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 this approach to identify some of them.