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.