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European Journal of Mathematics

, Volume 4, Issue 3, pp 1197–1263 | Cite as

Fano–Mukai fourfolds of genus 10 as compactifications of \({\mathbb {C}}^4\)

  • Yuri Prokhorov
  • Mikhail Zaidenberg
Research Article
  • 40 Downloads

Abstract

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 \({\mathbb {C}}^4\) in two different ways. Up to isomorphism, there is a unique fourfold \(V_{18}^{{\mathrm {s}}}\) acted upon by \({\mathrm{SL}}_2({\mathbb {C}})\). The group Open image in new window is a semidirect product Open image in new window . Furthermore, \(V_{18}^{{\mathrm {s}}}\) is a \({\mathrm{GL}}_2({\mathbb {C}})\)-equivariant completion of \({\mathbb {C}}^4\), and as well of \({\mathrm{GL}}_2({\mathbb {C}})\). The restriction of the \({\mathrm{GL}}_2({\mathbb {C}})\)-action on \(V_{18}^{{\mathrm {s}}}\) to Open image in new window yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold \(V_{18}^{\mathrm a}\) such that the group Open image in new window is a semidirect product Open image in new window . For a Fano–Mukai fourfold \(V_{18}\) isomorphic neither to \(V_{18}^{{\mathrm {s}}}\), nor to \(V_{18}^{\mathrm a}\), the group Open image in new window is a semidirect product of \(({{\mathbb {G}}}_{\mathrm {m}})^2\) and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety \(V_{18}\) is flexible in codimension one, and flexible if \(V_{18}=V_{18}^{{\mathrm {s}}}\).

Keywords

Fano variety Fourfold Compactification of \({\mathbb {C}}^n\) Sarkisov link Group action Automorphism group Affine cone 

Mathematics Subject Classification

14J35 14J45 14J50 14L30 14R10 14R20 

Notes

Acknowledgements

The paper started during the first author’s stay at the Institute Fourier, Grenoble, in June of 2016. He thanks the institute for its hospitality. The authors are grateful to Alexander Kuznetsov for useful discussions, to Michel Brion, Jun-Muk Hwang, and Laurent Manivel for important remarks around the material of Sect. 7, and to Ivan Arzhantsev and Alexander Perepechko for a pertinent remark concerning the material of Sect. 14. Our thanks are due also to the referee for his remarks improving the style of the paper.

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Copyright information

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Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Université Grenoble Alpes, CNRS, Institut FourierGrenobleFrance

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