Japanese Journal of Mathematics

, Volume 13, Issue 1, pp 109–185 | Cite as

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

  • Alexander G. Kuznetsov
  • Yuri G. Prokhorov
  • Constantin A. Shramov
Original Paper


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.

Mathematics Subject Classification (2010)

14J45 14J50 14J30 14C05 

Keywords and phrases

Fano variety Hilbert scheme automorphism group line conic derived category 


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Copyright information

© The Mathematical Society of Japan and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Alexander G. Kuznetsov
    • 1
    • 2
    • 3
  • Yuri G. Prokhorov
    • 1
    • 3
    • 4
  • Constantin A. Shramov
    • 1
    • 3
  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.The Poncelet LaboratoryIndependent University of MoscowMoscowRussia
  3. 3.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  4. 4.Department of AlgebraMoscow State UniversityMoscowRussia

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