A family of algebraic curves covering a projective variety is called a web of curves on if it has only finitely many members through a general point of . A web of curves on induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of . We study how the local differential geometry of the web-structure affects the global algebraic geometry of . 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 , up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when is a Fano submanifold of Picard number and the family of lines covering becomes a web. In this special case, we have the stronger result that the local differential geometry of the web-structure determines up to biregular equivalences. As an application, we show that if , , are two such Fano manifolds of Picard number , then any surjective morphism is an isomorphism.
"Geometry of webs of algebraic curves." Duke Math. J. 166 (3) 495 - 536, 15 February 2017. https://doi.org/10.1215/00127094-3715296