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Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on acomplex surface $M$, smooth along an irreducible component $D$ of itsdiscriminant $\Delta(\mathcal{W}),$ we establish an effective criterion for theholomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results ondecomposable webs due to Mar\'{\i}n, Pereira and Pirio. As an application, wededuce a complete characterization for the holomorphy of the curvature of theLegendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneousfoliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$generalizing some of our previous results. This then allows us to study theflatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case wherethe 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 Galoisgroup of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$is always flat.

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