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

Let $d\ge 3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $\mathrm{\Delta }(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Marín, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{L}\mathrm{e}\mathrm{g}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on ${\mathbb{P}}_{\mathbb{C}}^2,$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{L}\mathrm{e}\mathrm{g}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{L}\mathrm{e}\mathrm{g}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms ${\omega }_1^d=y^d\mathrm{d}x-x^d\mathrm{d}y$, ${\omega }_2^d=x^d\mathrm{d}x-y^d\mathrm{d}y\mathrm{.}$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{L}\mathrm{e}\mathrm{g}\mathcal{H}$ is always flat.