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Number of singularities of a generic web on the complex projective plane. (English) Zbl 1066.37032
Summary: Given a generic \(d\)-web \(\mathcal W_d\) of degree \(n\) in \(\mathbb C \mathbb P^2\), we associate with it a triple
\((S_{\mathcal W_d}, \pi|_{S_{\mathcal W_d}}, \mathcal F_{\mathcal W_d})\), where \(S_{\mathcal W_d}\) is a surface in \(\mathbb PT^* \mathbb C\mathbb P^2\), the projective cotangent bundle of \(\mathbb C \mathbb P^2\), \(\pi|_{S_{\mathcal W_d}}\) is the restriction of the natural projection \(\mathbb P T^*\mathbb C \mathbb P^2 \rightarrow \mathbb C \mathbb P^2\) to \(S_{\mathcal W_d}\) and \(\mathcal F_{\mathcal W_d}\) is a foliation on \(S_{W_d}\) 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 \(\mathcal F_{\mathcal W_d}\) in terms of \(d\) and \(n\). These results are compared with the case \(d=1\) (foliation in \(\mathbb C \mathbb P^2\)). We also calculate the total number of nodes and cusps of the projection \(\pi |_{S_{\mathcal W_d}}\) in terms of \(d\) and \(n\).

MSC:
37F75 Dynamical aspects of holomorphic foliations and vector fields
34M45 Ordinary differential equations on complex manifolds
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