Skip to main content
SpringerLink
Log in

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

  • Research Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

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 is a semidirect product . 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 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 is a semidirect product . For a Fano–Mukai fourfold \(V_{18}\) isomorphic neither to \(V_{18}^{{\mathrm {s}}}\), nor to \(V_{18}^{\mathrm a}\), the group 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 \(V_{18}=V_{18}^{{\mathrm {s}}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Cf. Lemma 5.4.

  2. Such cones are the only cubic cones in V, see 4.5. Hence in the sequel we call them simply cubic cones. We will show that any smooth variety \(V_{18}\) contains a cubis scroll or a cubic cone.

  3. In [41, Corollary 5.13] we constructed a pair (V, S) such that \(S\in {\mathscr {S}}(V)\) is a smooth cubic scroll.

  4. In fact, \(s_t\) is the exceptional section of a surface \(\widetilde{S}_t\cong \mathbb {F}_3\) in \({{\mathscr {L}}}(V)\). The restriction \(s|_{\widetilde{S}_t}:\widetilde{S}_t\rightarrow S_t\) is the minimal resolution of singularity of the cubic cone \(S_t\).

  5. The cubic cones \(S\subset V\) such that \((\Upsilon ,J_S)\) is of type 4.3 (ii) correspond actually to the two outer \((-1)\)-vertices in diagram (10.1.1). The two inner \((-1)\)-vertices in (10.1.1) correspond to the cubic cones \(S\subset V\) with \((\Upsilon ,J_S)\) of type 4.3 (iii); see Remark 13.3.2.

  6. Alternatively, one can notice that the cones \(S_1\) and \(S_2\) correspond to the disjoint exceptional sections of the components \(\mathscr {F}_1\) and \(\mathscr {F}_2\) of \(\Sigma _{\mathrm {s}}(V)\). It follows that \(S_1\cap S_2\) is zero-dimensional, hence empty since in \(H^*(V,\mathbb {Z})\), see Proposition 9.6 and Corollary 9.7.4.

References

  1. Akhiezer, D.N.: Lie Group Actions in Complex Analysis. Aspects of Mathematics, vol. E27. Friedr. Vieweg & Sohn, Braunschweig (1995)

  2. Andreatta, M., Wiśniewski, J.A.: On contractions of smooth varieties. J. Algebraic Geom. 7(2), 253–312 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Arzhantsev, I., Flenner, H., Kaliman, S., Kutzschebauch, F., Zaidenberg, M.: Flexible varieties and automorphism groups. Duke Math. J. 162(4), 767–823 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer, Berlin (2002) (Translated from the 1968 French original by Andrew Pressley)

  5. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Modern Birkhäuser Classics. Birkhäuser, Boston (2010) (Reprint of the 1997 edn.)

  6. Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold, New York (1993)

    MATH  Google Scholar 

  7. Coray, D.F., Tsfasman, M.A.: Arithmetic on singular Del Pezzo surfaces. Proc. London Math. Soc. (3) 57(1), 25–87 (1988)

  8. Demazure, M.: Automorphismes et déformations des variétés de Borel. Invent. Math. 39(2), 179–186 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolgachev, I.V.: Classical Algebraic Geometry. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  10. Eisenbud, D., Harris, J.: On varieties of minimal degree (A centennial account). In: Bloch, S.J. (ed.) Algebraic Geometry, Bowdoin, 1985. Proceedings of the Symposium Pure Mathematics, vol. 46.1, pp. 3–13. American Mathematical Society, Providence (1987)

  11. Eisenbud, D., Van de Ven, A.: On the normal bundles of smooth rational space curves. Math. Ann. 256(4), 453–463 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ekedahl, T., Laksov, D.: Two “generic” proofs of the spectral mapping theorem. Amer. Math. Monthly 111(7), 572–585 (2004)

  13. Fu, B., Hwang, J.-M.: Isotrivial VMRT-structures of complete intersection type. Asian J. Math., special issue dedicated to Ngaiming Mok’s 60th birthday (2016, to appear). arXiv:1608.00846

  14. Fujita, T.: On the structure of polarized manifolds with total deficiency one. II. J. Math. Soc. Japan 33(3), 415–434 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Furushima, M.: Complex analytic compactifications of \(\mathbf{C}^3\). Compositio Math. 76(1–2), 163–196 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Furushima, M.: The complete classification of compactifications of \(\mathbf{C}^3\) which are projective manifolds with the second Betti number one. Math. Ann. 297(4), 627–662 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994) (Reprint of the 1978 original)

  18. Hirzebruch, F.: Some problems on differentiable and complex manifolds. Ann. Math. 60, 213–236 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hochschild, G.: Introduction to Affine Algebraic Groups. Holden-Day, San Francisco (1971)

    MATH  Google Scholar 

  20. Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, New York (1975)

  21. Hwang, J.-M., Mok, N.: Deformation rigidity of the rational homogeneous space associated to a long simple root. Ann. Sci. Éc. Norm. Supér. (4) 35(2), 173–184 (2002)

  22. Iskovskikh, V.A.: Fano threefolds. I. Izv. Ross. Akad. Nauk SSSR Ser. Mat. 41(3), 516–562 (1977) (in Russian)

  23. Iskovskikh, V.A., Prokhorov, Yu.G.: Fano Varieties. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Geometry V. Encyclopaedia of Mathematical Sciences, vol. 47. Springer, Berlin (1999)

  24. Kapustka, M., Ranestad, K.: Vector bundles on Fano varieties of genus ten. Math. Ann. 356(2), 439–467 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: \(\mathbb{G}_{{\rm a}}\)-actions on affine cones. Transform. Groups 18(4), 1137–1153 (2013)

  26. Kodaira, K.: Holomorphic mappings of polydiscs into compact complex manifolds. J. Differential Geom. 6, 33–46 (1971/72)

  27. Kurtzke Jr., J.F.: Centralizers of irregular elements in reductive algebraic groups. Pacific J. Math. 104(1), 133–154 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuznetsov, A.G., Prokhorov, YuG, Shramov, C.A.: Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Japan J. Math. 13(1), 109–185 (2018)

  29. Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(1), 65–100 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. Nagoya Math. J. 11, 145–150 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  31. Michałek, M., Perepechko, A., Süss, H.: Flexible affine cones and flexible coverings. Math. Z. (2016, to appear). arXiv:1612.01144v1

  32. Mostow, G.D.: Fully reducible subgroups of algebraic groups. Amer. J. Math. 78, 200–221 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mukai, Sh.: Curves, \(K3\) surfaces and Fano \(3\)-folds of genus \(\leqslant 10\). In: Hijikata, H., et al. (eds.) Algebraic Geometry and Commutative Algebra, vol. I, pp. 357–377. Kinokuniya, Tokyo (1988)

  34. Mukai, Sh.: Biregular classification of Fano \(3\)-folds and Fano manifolds of coindex \(3\). Proc. Nat. Acad. Sci. USA 86(9), 3000–3002 (1989)

  35. Peternell, T., Schneider, M.: Compactifications of \(\mathbf{C}^3\). I. Math. Ann. 280(1), 129–146 (1988)

    Article  MATH  Google Scholar 

  36. Piontkowski, J., Van de Ven, A.: The automorphism group of linear sections of the Grassmannians \({\mathbb{G}} (1, N)\). Doc. Math. 4, 623–664 (1999)

    MathSciNet  MATH  Google Scholar 

  37. Prokhorov, YuG: Fano threefolds of genus \(12\) and compactifications of \(\mathbb{C}^3\). St. Petersburg Math. J. 3(4), 855–864 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Prokhorov, YuG: Compactifications of \(\mathbf{C}^4\) of index \(3\). In: Tikhomirov, A., Tyurin, A. (eds.) Algebraic Geometry and its Applications (Yaroslavl, 1992). Aspects of Mathematics, vol. E25, pp. 159–169. Vieweg, Braunschweig (1994)

    Chapter  Google Scholar 

  39. Prokhorov, YuG: On \(G\)-Fano threefolds. Izv. Math. 79(4), 795–808 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Prokhorov, YuG: Singular Fano threefolds of genus \(12\). Sb. Math. 207(7), 983–1009 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Prokhorov, Yu., Zaidenberg, M.: New examples of cylindrical Fano fourfolds. In: Masuda, K., et al. (eds.) Algebraic Varieties and Automorphism Groups. Advanced Studies in Pure Mathematics, vol. 75, pp. 443–464. Mathematical Society of Japan, Tokyo (2017)

    Google Scholar 

  42. Prokhorov, Yu., Zaidenberg, M.: Examples of cylindrical Fano fourfolds. Eur. J. Math. 2(1), 262–282 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Shokurov, V.V.: The existence of a straight line on Fano \(3\)-folds. Math. USSR-Izv. 15(1), 173–209 (1980)

    Article  MATH  Google Scholar 

  44. Springer, T.A., Steinberg, R.: Conjugacy classes. Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Mathematics, vol. 131, pp. 167–266. Springer, Berlin (1970)

  45. Steinberg, R.: Regular elements of semisimple algebraic groups. Publ. Math. Inst. Hautes Études Sci. 25, 49–80 (1965)

    Article  MATH  Google Scholar 

  46. Tevelev, E.A.: Projective Duality and Homogeneous Spaces. Encyclopaedia of Mathematical Sciences, vol. 133. Springer, Berlin (2005)

  47. Tian, G.: Kähler-Einstein metrics on Fano manifolds. Japan J. Math. 10(1), 1–41 (2015)

    Article  MATH  Google Scholar 

  48. Todd, J.A.: The locus representing the lines of four-dimensional space and its application to linear complexes in four dimensions. Proc. London Math. Soc. S2–30, 513–550 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zaidenberg, M.G.: An analytic cancellation theorem and exotic algebraic structures on \(\mathbb{C}^n\), \(n\geqslant 3\). Colloque d’analyse complexe et géométrie. Astérisque 217(8), 251–282 (1993)

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, 8 Gubkina street, 119991, Moscow, Russia

    Yuri Prokhorov

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobevy Gory, GSP-1, 119991, Moscow, Russia

    Yuri Prokhorov

  3. National Research University Higher School of Economics, 6 Usacheva Street, 119048, Moscow, Russia

    Yuri Prokhorov

  4. Université Grenoble Alpes, CNRS, Institut Fourier, 38000, Grenoble, France

    Mikhail Zaidenberg

Authors
  1. Yuri Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Zaidenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Prokhorov.

Additional information

Yuri Prokhorov was partially supported by the RFBR Grants 15-01-02164, 15-01-02158, and by the Russian Academic Excellence Project ‘5-100’.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prokhorov, Y., Zaidenberg, M. Fano–Mukai fourfolds of genus 10 as compactifications of \({\mathbb {C}}^4\) . European Journal of Mathematics 4, 1197–1263 (2018). https://doi.org/10.1007/s40879-018-0244-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-018-0244-y

Keywords

Mathematics Subject Classification

Search

Navigation