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⊂PN 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:X→X' is an isomorphism.