Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 17, Issue 1

Issues

Connectedness Bertini Theorem via numerical equivalence

Diletta Martinelli
  • Corresponding author
  • James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Juan Carlos Naranjo / Gian Pietro Pirola
Published Online: 2017-01-08 | DOI: https://doi.org/10.1515/advgeom-2016-0028

Abstract

Let X be an irreducible projective variety and let f : X → ℙn be a morphism. We give a new proof of the fact that the preimage of any linear variety of dimension kn + 1 -dim f(X) is connected. We show that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as an application of our main theorem.

Keywords: Bertini Theorem; connectedness; numerical equivalence; Hodge Index Theorem

MSC 2010: 14J70; 14J99

Communicated by: A. Sommese

References

  • [1]

    E. Bertini, Sui sistemi lineari. Ist. Lombardo, Rend., II. Ser. 15 (1882), 24-29. Zbl 14.0433.02Google Scholar

  • [2]

    A. J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. no. 83 (1996), 51-93. MR1423020 Zbl 0916.14005Google Scholar

  • [3]

    P. Deligne, Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton). In: Bourbaki Seminar, Vol. 1979/80, volume 842 of Lecture Notes in Math., 1–10, Springer 1981. MR636513 Zbl 0478.14008Google Scholar

  • [4]

    W. Fulton, J. Hansen, A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings. Ann. of Math. (2) 110 (1979), 159–166. MR541334 Zbl 0389.14002Google Scholar

  • [5]

    W. Fulton, R. Lazarsfeld, Connectivity and its applications in algebraic geometry. In: Algebraic geometry (Chicago, Ill., 1980), volume 862 of Lecture Notes in Math., 26–92, Springer 1981. MR644817 Zbl 0484.14005Google Scholar

  • [6]

    J. Hansen, A connectedness theorem for flagmanifolds and Grassmannians. Amer. J. Math. 105 (1983), 633–639. MR704218 Zbl 0544.14034Google Scholar

  • [7]

    R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.14001Google Scholar

  • [8]

    J.-P. Jouanolou, Théorèmes de Bertini et applications. Birkhäuser 1983. MR725671 Zbl 0519.14002Google Scholar

  • [9]

    S. L. Kleiman, Bertini and his two fundamental theorems. Rend. Circ. Mat. Palermo (2) Suppl. no. 55 (1998), 9–37. MR1661859 Zbl 0926.14001Google Scholar

  • [10]

    J. Kollár, editor, Complex algebraic geometry, volume 3 of IAS/Park City Mathematics Series. Amer. Math. Soc. 1997. MR1442521 Zbl 0866.00043Google Scholar

  • [11]

    R. Lazarsfeld, Positivity in algebraic geometry. Springer 2004. MR2095471 Zbl 1093.14501 Zbl 1066.14021Google Scholar

  • [12]

    A. Seidenberg, The hyperplane sections of normal varieties. Trans. Amer. Math. Soc. 69 (1950), 357–386. MR0037548 Zbl 0040.23501Google Scholar

  • [13]

    A. J. Sommese, A. Van de Ven, Homotopy groups of pullbacks of varieties. Nagoya Math. J. 102 (1986), 79–90. MR846130 Zbl 0564.14010Google Scholar

  • [14]

    A. Weil, Foundations of algebraic geometry. Amer. Math. Soc. 1962. MR0144898 Zbl 0168.18701Google Scholar

  • [15]

    F. L. Zak, Tangents and secants of algebraic varieties, volume 127 of Translations of Mathematical Monographs. Amer. Math. Soc. 1993. MR1234494 Zbl 0795.14018Google Scholar

About the article

Received: 2014-12-15

Published Online: 2017-01-08

Published in Print: 2017-01-01


Citation Information: Advances in Geometry, Volume 17, Issue 1, Pages 31–38, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0028.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in