## Duke Mathematical Journal

### Geometry of webs of algebraic curves

Jun-Muk Hwang

#### 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 $X\subset\mathbb{P}^{N}$ 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 $X,X'\subset\mathbb{P}^{N}$, $\operatorname{dim}X'\geq3$, are two such Fano manifolds of Picard number $1$, then any surjective morphism $f:X\to X'$ 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

#### 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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