Geometry of webs of algebraic curves

TL;DR

This paper investigates how the local differential geometry of webs of algebraic curves on projective varieties influences the global algebraic structure, establishing conditions under which local data determines the entire variety.

Contribution

It proves that under specific geometric conditions, the local differential geometry of webs of curves uniquely determines the global algebraic geometry of the variety.

Findings

Local differential geometry determines the algebraic structure of the variety.

Conditions like pairwise non-integrability and bracket-generating are crucial.

For certain Fano manifolds, the geometry determines the variety up to isomorphism.

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 non-integrability 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 $X \subset {\bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X' \subset {\bf P}^N, \dim X' \geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X \to X'$ is an isomorphism.