## Duke Mathematical Journal

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

### Geometry of webs of algebraic curves

#### 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 $X,X\text{'}\subset {\mathbb{P}}^{N}$, $dimX\text{'}\ge 3$, are two such Fano manifolds of Picard number $1$, then any surjective morphism $f:X\to X\text{'}$ 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