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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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