Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Special birational transformations of type $ (2,1)$

Authors: Baohua Fu and Jun-Muk Hwang
Journal: J. Algebraic Geom. 27 (2018), 55-89
Published electronically: March 10, 2017
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Abstract | References | Additional Information

Abstract: A birational transformation $ \Phi : \mathbb{P}^n \dasharrow Z \subset \mathbb{P}^N,$ where $ Z \subset \mathbb{P}^N$ is
a nonsingular variety of Picard number 1, is called a special birational transformation of type $ (a,b)$ if $ \Phi $ is given by a linear system of
degree $ a$, its inverse $ \Phi ^{-1}$ is given by a linear system of degree $ b$ and the base locus $ S \subset \mathbb{P}^n$ of $ \Phi $ is irreducible and nonsingular. In this paper, we classify special birational transformations of type $ (2,1)$. In addition to previous works by Alzati-Sierra and Russo on this topic, our proof employs natural $ \mathbb{C}^*$-actions on $ Z$ in a crucial way. These $ \mathbb{C}^*$-actions also relate our result to the prolongation problem studied in our previous work.

References [Enhancements On Off] (What's this?)

  • [AS] Alberto Alzati and José Carlos Sierra, Quadro-quadric special birational transformations of projective spaces, Int. Math. Res. Not. IMRN 1 (2015), 55-77. MR 3340294
  • [ES] Lawrence Ein and Nicholas Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), no. 5, 783-800. MR 1020829,
  • [FH] Baohua Fu and Jun-Muk Hwang, Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity, Invent. Math. 189 (2012), no. 2, 457-513. MR 2947549,
  • [GH] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
  • [Ha] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • [HM05] Jun-Muk Hwang and Ngaiming Mok, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math. 160 (2005), no. 3, 591-645. MR 2178704,
  • [IR1] Paltin Ionescu and Francesco Russo, Conic-connected manifolds, J. Reine Angew. Math. 644 (2010), 145-157. MR 2671777,
  • [IR2] Paltin Ionescu and Francesco Russo, Varieties with quadratic entry locus. II, Compos. Math. 144 (2008), no. 4, 949-962. MR 2441252,
  • [IP] V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1-247. MR 1668579
  • [IL] Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, RI, 2003. MR 2003610
  • [L] J. M. Landsberg, On second fundamental forms of projective varieties, Invent. Math. 117 (1994), no. 2, 303-315. MR 1273267,
  • [Mo] Ngaiming Mok, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, Third International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt.  1, vol. 2, Amer. Math. Soc., Providence, RI, 2008, pp. 41-61. MR 2409622
  • [M] Shigeru Mukai, Biregular classification of Fano $ 3$-folds and Fano manifolds of coindex $ 3$, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000-3002. MR 995400,
  • [NO] Carla Novelli and Gianluca Occhetta, Projective manifolds containing a large linear subspace with nef normal bundle, Michigan Math. J. 60 (2011), no. 2, 441-462. MR 2825270,
  • [P] Boris Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963-987. MR 2507635,
  • [R1] Francesco Russo, On a theorem of Severi, Math. Ann. 316 (2000), no. 1, 1-17. MR 1735076,
  • [R2] Francesco Russo, Varieties with quadratic entry locus. I, Math. Ann. 344 (2009), no. 3, 597-617. MR 2501303,
  • [S] Eiichi Sato, Projective manifolds swept out by large-dimensional linear spaces, Tohoku Math. J. (2) 49 (1997), no. 3, 299-321. MR 1464179,
  • [Z] F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494

Additional Information

Baohua Fu
Affiliation: Institute of Mathematics, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun East Road, Beijing, 100190, People’s Republic of China – and – School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, People’s Republic of China

Jun-Muk Hwang
Affiliation: Korea Institute for Advanced Study, Hoegiro 85, Seoul, 130-722, Republic of Korea

Received by editor(s): September 21, 2015
Received by editor(s) in revised form: May 4, 2016
Published electronically: March 10, 2017
Additional Notes: The first author was supported by the National Natural Science Foundation of China (11225106 and 11321101). The second author was supported by the National Researcher Program 2010-0020413 of the NRF
Article copyright: © Copyright 2017 University Press, Inc.

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