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

Abstract

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.

Maps from K-trivial varieties and connectedness problems

“…By construction, we obtain a rational dominant map h : Z H, hence H is rational. This contradicts the classification of H, see [KPS18].…”

‐subgroups in the space Cremona group

“…Lemma (Cf. [, Lemma 5.3], [, Lemma 4.4.1], [, Proposition 6.1.1]). Let X be a Fano threefold with canonical Gorenstein singularities, and let be a p ‐group.…”

“…Thus and X is smooth by [, Theorem 1.3]. Hence the family parameterizing the conics (in the anticanonical embedding) on X is isomorphic to the projective plane , see [, Theorem 2.4], [, Proposition B.4.1]. By Lemma the group G has a fixed point on , which means that there is a G ‐invariant conic contained in X .…”

S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

“…Note, however, that the classification of Fano 3‐folds with ‐actions and Picard rank greater than 1 was not yet carried out. The case of Picard rank 1 was settled in .…”