Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{ W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $\Delta(\mathcal{ W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{ W}$ along $D,$ generalizing results on decomposable webs due to Mar\'{ \i}n, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{ Leg}\mathcal{ H}$ of a homogeneous foliation $\mathcal{ H}$ of degree $d$ on $\mathbb{ P}^{ 2}_{ \mathbb{ C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{ Leg}\mathcal{ H}$ in the particular case where the foliation $\mathcal{ H}$ is Galois. When the Galois group of $\mathcal{ H}$ is cyclic, we show that $\mathrm{ Leg}\mathcal{ H}$ is flat if and only if $\mathcal{ H}$ is given, up to linear conjugation, by one of the two 1-forms $\omega_1^{ \hspace{ 0.2mm}d}=y^d\mathrm{ d}x-x^d\mathrm{ d}y$, $\omega_2^{ \hspace{ 0.2mm}d}=x^d\mathrm{ d}x-y^d\mathrm{ d}y.$ When the Galois group of $\mathcal{ H}$ is non-cyclic, we obtain that $\mathrm{ Leg}\mathcal{ H}$ is always flat.
Categories: math.DS math.CA math.CV math.DG