Preprint C20/2003
Generic Webs On the Complex Projective Plane
Joseph Nee Anyah Yartey
Keywords: Webs | cusps | nodes and folds
Given a generic d-web Wd of degree n in CP(2), we associate to it a triple $(S_Wd; \pi|_{S_{W_d}},\Cal{F}(\alpha))$, where $S_{W_d}$ is a surface in $PT^*CP(2)$ ; the projective cotangent bundle of $CP(2)$ ; $\pi|_{S_{W_d}}$ is the restriction of the natural projection $PT^*CP(2)\to CP(2)$ to $S_{W_d}$ and $\Cal{F}(\alpha)$ is a foliation on $S_{W_d}}$ given by a special meromorphic 1-form $\alpha$. In this thesis we give the exact form of a d-web of degree n in CP(2) and conditions under which it is generic. Hence given a generic d-web of degree n in CP(2), we calculate the total number of singularities and the sum of the indices of Baum-Bott for the foliation $\Cal{F}(\alpha))$ in terms of d and n. The results are compared with the case d=1. We also calculate the total number of nodes and cusps of the projection $\pi|_{S_{W_d}}$ in terms of d and n.