Ordinary holomorphic webs of codimension one

Vincent Cavalier, Daniel Lehmann

Published 2007-03-20, updated 2008-10-13Version 2

The main change with respect to the previous version is a change of terminology : we call "ordinary" the webs previously called "regular". A holomorphic $d$-web of codimension one in dimension $n$ is "ordinary", if it satisfies to some condition of genericity. In dimension at least 3, any such web has a rank bounded from above by a number $\pi'(n,d)$ strictly smaller than the bound $\pi(n,d)$ of castelnuovo. This bound $\pi'(n,d)$ is optimal. Moreover, for some $d$'s, the abelian relations are sections with vanishing covariant derivative of some bundle with a connection, the curvature of which generalizes the Blaschke curvature. In dimension 2, we recover results of H\'enaut and Pantazi