Duke Mathematical Journal

Geometry of webs of algebraic curves

Jun-Muk Hwang

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1475602128

Digital Object Identifier
doi:10.1215/00127094-3715296

Mathematical Reviews number (MathSciNet)
MR3606724

Zentralblatt MATH identifier
1372.14043

Subjects
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]

Keywords
web geometry extension of holomorphic maps minimal rational curves Fano varieties

Citation

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


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