Osaka Journal of Mathematics

Affine cones over Fano threefolds and additive group actions

Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg

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In this paper we address the following questions for smooth Fano threefolds of Picard number 1: \begin{itemize} \item \textit{When does such a threefold $X$ possess an open cylinder $U \simeq Z\times\mathbb{A}^{1}$, where $Z$ is a surface?} \item \textit{When does an affine cone over $X$ admit an effective action of the additive group of the base field?} \end{itemize} A geometric criterion from [26] (see also [27]) says that the two questions above are equivalent. In [26] we found some interesting families of Fano threefolds carrying a cylinder. Here we provide new such examples.

Article information

Osaka J. Math., Volume 51, Number 4 (2014), 1093-1113.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30] 14J45: Fano varieties
Secondary: 14J50: Automorphisms of surfaces and higher-dimensional varieties 14R05: Classification of affine varieties


Kishimoto, Takashi; Prokhorov, Yuri; Zaidenberg, Mikhail. Affine cones over Fano threefolds and additive group actions. Osaka J. Math. 51 (2014), no. 4, 1093--1113.

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  • I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch and M. Zaidenberg: Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), 767–823.
  • A. Beauville: Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), 309–391.
  • J. Blanc and S. Lamy: Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, Proc. Lond. Math. Soc. (3) 105 (2012), 1047–1075.
  • \begingroup F. Bogomolov, I. Karzhemanov and K. Kuyumzhiyan: Unirationality and existence of infinitely transitive models; in Birational Geometry, Rational Curves, and Arithmetic, Springer, New York, 2013, 77–92. \endgroup
  • C.H. Clemens and P.A. Griffiths: The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.
  • S.D. Cutkosky: On Fano $3$-folds, Manuscripta Math. 64 (1989), 189–204.
  • I.V. Dolgachev: Classical Algebraic Geometry, Cambridge Univ. Press, Cambridge, 2012.
  • T. Fujita: On the structure of polarized manifolds with total deficiency one, II, J. Math. Soc. Japan 33 (1981), 415–434.
  • M. Furushima: The complete classification of compactifications of $\mathbf{C}^{3}$ which are projective manifolds with the second Betti number one, Math. Ann. 297 (1993), 627–662.
  • P. Griffiths and J. Harris: Principles of Algebraic Geometry, reprint of the 1978 original, Wiley, New York, 1994.
  • M.M. Grinenko: Mori structures on a Fano threefold of index 2 and degree 1, Proc. Steklov Inst. Math. 246 (2004), 103–128.
  • L. Gruson, F. Laytimi and D.S. Nagaraj: On prime Fano threefolds of genus 9, Internat. J. Math. 17 (2006), 253–261.
  • R. Hartshorne: Algebraic Geometry, Springer, New York, 1977.
  • F. Hidaka and K. Watanabe: Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4 (1981), 319–330.
  • F. Hirzebruch: Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60 (1954), 213–236.
  • A. Iliev: The $\mathrm{Sp}_{3}$-Grassmannian and duality for prime Fano threefolds of genus 9, Manuscripta Math. 112 (2003), 29–53.
  • V.A. Iskovskikh: Fano threefolds, I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516–562.
  • V.A. Iskovskikh: Double projection from a line onto Fano $3$-folds of the first kind, Math. USSR-Sb. 66 (1990), 265–284.
  • V.A. Iskovskikh: Anticanonical models of three-dimensional algebraic varieties, J. Sov. Math. 13 (1980), 745–814.
  • V.A. Iskovskikh: Birational automorphisms of three-dimensional algebraic varieties, J. Sov. Math. 13 (1980), 815–868.
  • \begingroup V.A. Iskovskikh and Yu.I. Manin: Three-dimensional quartics and counterexamples to the Lüroth problem, Math. USSR Sb. 15 (1971), 141–166 (1972). \endgroup
  • V.A. Iskovskikh and Yu.G. Prokhorov: Fano varieties; in Algebraic Geometry, V, Encyclopaedia Math. Sci. 47, Springer, Berlin, 1999, 1–247.
  • V.A. Iskovskikh and A.V. Pukhlikov: Birational automorphisms of multidimensional algebraic manifolds, J. Math. Sci. 82 (1996), 3528–3613.
  • T. Kambayashi and M. Miyanishi: On flat fibrations by the affine line, Illinois J. Math. 22 (1978), 662–671.
  • T. Kambayashi and D. Wright: Flat families of affine lines are affine-line bundles, Illinois J. Math. 29 (1985), 672–681.
  • T. Kishimoto, Yu. Prokhorov and M. Zaidenberg: Group actions on affine cones; in Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, RI, 2011, 123–163.
  • \begingroup T. Kishimoto, Yu. Prokhorov and M. Zaidenberg: $\mathbb{G}_{\mathrm{a}}$-actions on affine cones, Transform. Groups, 18 (2013), 1137–1153. \endgroup
  • T. Kishimoto, Yu. Prokhorov and M. Zaidenberg: Unipotent group actions on del Pezzo cones, Algebr. Geom. 1 (2014), 46–56.
  • J. Kollár: Flops, Nagoya Math. J. 113 (1989), 15–36.
  • Y. Miyaoka and S. Mori: A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), 65–69.
  • S. Mori: Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), 133–176.
  • M. Nagata: On rational surfaces, I. Irreducible curves of arithmetic genus $0$ or $1$, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351–370.
  • A.Yu. Perepechko: Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5, Funktsional. Anai. i Prilozhen. 47 (2013), 45–52, (in Russian).
  • Yu. Prokhorov, Geometrical properties of Fano threefolds, PhD thesis, Moscow State Univ. 1990, (in Russian).
  • Yu. Prokhorov: Exotic Fano varieties, Moscow Univ. Math. Bull. 45 (1990), 36–38.
  • Yu. Prokhorov: Automorphism groups of Fano $3$-folds, Russian Math. Surveys 45 (1990), 222–223.
  • M. Reid: Lines on Fano $3$-folds according to Shokurov, Technical Report 11, Mittag-Leffler Inst., 1980.
  • M. Reid: Minimal models of canonical $3$-folds; in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983, 131–180.
  • M. Reid: Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), 695–727.
  • V.V. Šokurov: The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 922–964.
  • K. Takeuchi: Some birational maps of Fano $3$-folds, Compositio Math. 71 (1989), 265–283.
  • A.N. Tyurin: The middle Jacobian of three-dimensional varieties, J. Sov. Math. 13 (1980), 707–745.
  • C. Voisin: Sur la jacobienne intermédiaire du double solide d'indice deux, Duke Math. J. 57 (1988), 629–646.