Number of Singularities of a Generic Web on the Complex Projective Plane

Abstract

Given a generic d-web Wd of degree n in CP2, we associate with it a triple (SWd, π|SWd ,FWd ), where SWd is a surface in PT∗CP2, the projective cotangent bundle of CP2, π|SWd is the restriction of the natural projection PT∗CP2 −→ CP2 to SWd and FWd is a foliation on SWd given by a special meromorphic 1-form. The main objective of this article is to calculate the total number of singularities and the sum of the indices of Baum–Bott for the foliation FWd in terms of d and n. These results are compared with the case d = 1 (foliation in CP2). We also calculate the total number of nodes and cusps of the projection π|SWd in terms of d and n.