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15 February 2017 Geometry of webs of algebraic curves
Jun-Muk Hwang
Duke Math. J. 166(3): 495-536 (15 February 2017). DOI: 10.1215/00127094-3715296

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.

Citation

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Jun-Muk Hwang. "Geometry of webs of algebraic curves." Duke Math. J. 166 (3) 495 - 536, 15 February 2017. https://doi.org/10.1215/00127094-3715296

Information

Received: 22 January 2015; Revised: 5 April 2016; Published: 15 February 2017
First available in Project Euclid: 4 October 2016

zbMATH: 1372.14043
MathSciNet: MR3606724
Digital Object Identifier: 10.1215/00127094-3715296

Subjects:
Primary: 14M22
Secondary: 14J45, 32D15, 32H04, 53A60

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 3 • 15 February 2017
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