On the holomorphy of the curvature of planar webs along an invariant curve

Let $\mathcal{W}=\mathcal{W}_{n}\boxtimes\mathcal{W}_{d-n}$ be a $d$-web on $(\mathbb{C}^2,0)$, where $\mathcal{W}_n$ is an $n$-web with a totally invariant irreducible curve~$C$, and $\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\mathcal{W}_n$ is holomorphic along $C$. When $\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\mathcal{W}_n)$, and hence $K(\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\mathcal{W}_n$ is irreducible and $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1),$ then $K(\mathcal{W})$ is holomorphic along $C.$ This generalizes a result of \textsc{Mar\'{\i}n} and \textsc{Pereira}, obtained in the case where $C$ has minimal multiplicity $n-1$ in the discriminant $\Delta(\mathcal{W}_n).$ If $n$ is prime or $n=4$, the condition $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1)$ can be weakened to $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<n(n-1).$ Moreover, we describe a natural decomposition of $\mathcal{W}_n$ as the product of two subwebs $\mathcal{W}_n=\mathcal{W}_{n}^{\rm{str}}\boxtimes\mathcal{W}_{n}^{\rm{wk}}.$ Under the assumption that $\mathcal{W}_{n}^{\rm{wk}}$ is non-degenerate along $C$, we show that the holomorphy of $K(\mathcal{W})$ on $C$ is equivalent to that of $K(\mathcal{W}_{n}^{\rm{str}}).$