• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Working paper

Fano-Mukai fourfolds of genus 10 as compactifications of ℂ^4

Prokhorov Y., Zaidenberg M.
It is known that the moduli space of smooth Fano-Mukai fourfolds V 18 of genus 10 has dimension one. We show that any such fourfold is a completion of C 4 in two different ways. Up to isomorphism, there is a unique fourfold V s 18 acted upon by SL 2 ( C ). The group Aut( V s 18 ) is a semidirect product GL 2 ( C ) o ( Z / 2 Z ). Furthermore, V s 18 is a GL 2 ( C )-equivariant completion of C 4 , and as well of GL 2 ( C ). The restriction of the GL 2 ( C )-action on V s 18 to C 4 ↪ → V s 18 yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold V a 18 such that the group Aut( V a 18 ) is a semidirect product ( G a × G m ) o ( Z / 2 Z ). For a Fano-Mukai fourfold V 18 neither isomorphic to V s 18 , nor to V a 18 , one has Aut 0 ( V 18 ) ∼ = ( G m ) 2 , and Aut( V 18 ) is a semidirect product of Aut 0 ( V 18 ) and a finite cyclic group whose order is a factor of 6.