A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on \(\mathbb{P}^2_{\mathbb{C}}\) (Q6648882)

The authors consider holomorphic \(d\)-webs \(W\) on complex surfaces for \(d>2\). In Theorem 2.1 they obtain a criterion for holomorphy of the curvature of a web with some special choice of local coordinates, namely the web is considered on some irreducible component \(y=0\) of the discriminant.\N\NLet \(H\) be a homogeneous foliation of degree \(d\) on the projective plane, \(\text{Leg}H\) be a \(d\)-web on the dual plane obtained from \(H\) by Legendre transformation. In Theorem 3.1, the authors find a necessary condition for the holomorphy of the curvature of the web \(\text{Leg} H\). They consider in detail the case when \(H\) be a Galois homogeneous foliation and give some examples illustrating the main results of the paper.