Duke Mathematical Journal
- Duke Math. J.
- Volume 166, Number 3 (2017), 495-536.
Geometry of webs of algebraic curves
A family of algebraic curves covering a projective variety is called a web of curves on if it has only finitely many members through a general point of . A web of curves on induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of . We study how the local differential geometry of the web-structure affects the global algebraic geometry of . 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 , up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when is a Fano submanifold of Picard number and the family of lines covering becomes a web. In this special case, we have the stronger result that the local differential geometry of the web-structure determines up to biregular equivalences. As an application, we show that if , , are two such Fano manifolds of Picard number , then any surjective morphism is an isomorphism.
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]
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