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 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.

60 pages, LaTeX, final version, prepared for publication, to appear in Japanese Journal of Mathematics, section 5 is essentially rewritten

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Cited by in corpus (36)

• Stability conditions on Kuznetsov components
• The rationality problem for conic bundles
• Fano threefolds with infinite automorphism groups
• Fano threefolds as equivariant compactifications of the vector group
• A note on the Kuznetsov component of the Veronese double cone
• Automorphisms of weighted complete intersections
• Categorical Torelli theorems for Gushel-Mukai threefolds
• Delta-invariants for Fano varieties with large automorphism groups
• $S^1$-invariant symplectic hypersurfaces in dimension $6$ and the Fano condition
• Connected algebraic groups acting on three-dimensional Mori fibrations
• A desingularization of Kontsevich's compactification of twisted cubics in $V_5$
• Even and odd instanton bundles on Fano threefolds
• Kaehler-Einstein Fano threefolds of degree 22
• Bridgeland stability of minimal instanton bundles on Fano threefolds
• Polyvector fields for Fano 3-folds
• p-subgroups in the space Cremona group
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• Affine cones over Fano-Mukai fourfolds of genus 10 are flexible
• Zero-cycles on self-products of surfaces: some new examples verifying Voisin's conjecture
• Jordan constant for Cremona group of rank 3
• Rational curves on $V_5$ and rational simple connectedness
• Fano threefolds
• Finite groups of birational transformations
• On exceptional sets in Manin's Conjecture
• Fano-Mukai fourfolds of genus $10$ as compactifications of ${\mathbb{C}}^4$
• Maps from K-trivial varieties and connectedness problems
• Fano 3-folds from homogeneous vector bundles over Grassmannians
• Fano manifolds of coindex three admitting nef tangent bundle
• Fano 4-folds having a prime divisor of Picard number 1
• Rational curves on prime Fano threefolds of index 1
• Topology of spaces of regular sections and applications to automorphism groups
• Cylinders in Mori Fiber Spaces: forms of the quintic del Pezzo threefold
• On 3-folds having a holomorphic torus action with 6 fixed points
• Moduli spaces of quasi-trivial sheaves
• Smooth components on special iterated Hilbert schemes
• SYZ for index 1 Fano hypersurfaces in projective space