Duke Mathematical Journal

Geometry of webs of algebraic curves

Jun-Muk Hwang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A family of algebraic curves covering a projective variety X is called a web of curves on X if it has only finitely many members through a general point of X. A web of curves on X induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of X. We study how the local differential geometry of the web-structure affects the global algebraic geometry of X. Under two geometric assumptions on the web-structure—the pairwise nonintegrability condition and the bracket-generating condition—we prove that the local differential geometry determines the global algebraic geometry of X, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when XPN is a Fano submanifold of Picard number 1 and the family of lines covering X becomes a web. In this special case, we have the stronger result that the local differential geometry of the web-structure determines X up to biregular equivalences. As an application, we show that if X,X'PN, dimX'3, are two such Fano manifolds of Picard number 1, then any surjective morphism f:XX' is an isomorphism.

Article information

Duke Math. J., Volume 166, Number 3 (2017), 495-536.

Received: 22 January 2015
Revised: 5 April 2016
First available in Project Euclid: 4 October 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M22: Rationally connected varieties
Secondary: 32D15: Continuation of analytic objects 14J45: Fano varieties 32H04: Meromorphic mappings 53A60: Geometry of webs [See also 14C21, 20N05]

web geometry extension of holomorphic maps minimal rational curves Fano varieties


Hwang, Jun-Muk. Geometry of webs of algebraic curves. Duke Math. J. 166 (2017), no. 3, 495--536. doi:10.1215/00127094-3715296. https://projecteuclid.org/euclid.dmj/1475602128

Export citation


  • [1] J.-P. Demailly, “Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials” in Algebraic Geometry—Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, 1997, 285–360.
  • [2] J.-M. Hwang, “Geometry of minimal rational curves on Fano manifolds” in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335–393.
  • [3] J.-M. Hwang and N. Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number $1$, J. Math. Pures Appl. (9) 80 (2001), 563–575.
  • [4] J.-M. Hwang and N. Mok, Finite morphisms onto Fano manifolds of Picard number $1$ which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), 627–651.
  • [5] V. A. Iskovskikh and Yu. G. Prokhorov, Fano Varieties, Encyclopaedia Math. Sci. 47, Springer, Berlin, 1999.
  • [6] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • [7] J. V. Pereira and L. Pirio, An Invitation to Web Geometry, Publ. Mat. IMPA, IMPA, Rio de Janeiro, 2009.
  • [8] C. Schuhmann, Morphisms between Fano threefolds, J. Algebraic Geom. 8 (1999), 221–244.