**Proof. From (1 ) and (2 ), we obtain \[\begin{align} \prod\limits_{j=1}^{\nu}(\zeta^{jr}c_{0}(z)w^{\frac{r}{\nu}}+\ldots)=(-1)^{r(\nu+1)}w^{r}c_{0}(z)^{\nu}+\ldots=w^{\kappa}a_{0}(z,w). \end{align}\] Since \(a_0(z,0)\neq0,\) we deduce that \(r=\kappa\). Thus we have \[\begin{align} \Delta(\mathcal{W}_{\nu}) &=\prod\limits_{i\neq j}\Big((\zeta^{i\kappa}c_{0}(z)w^{\frac{\kappa}{\nu}}+\cdots)-(\zeta^{j\kappa}c_{0}(z)w^{\frac{\kappa}{\nu}}+\cdots)\Big)\\ &=w^{\kappa(\nu-1)}c_{0}(z)^{\nu(\nu-1)}\prod\limits_{i\neq j}(\zeta^{i\kappa}-\zeta^{j\kappa})+\cdots. \end{align}\] It follows that \(\mathrm{mult}\left(\Delta(\mathcal{W}_{\nu}),C\right)\geq\kappa(\nu-1),\) with equality if and only if \(\zeta^{i\kappa}\neq\zeta^{j\kappa}\) for all \(i\neq j\), that is, if and only if \(\gcd(\nu,\kappa)=1\), since \(\gcd(\nu,\kappa)=\dfrac{\nu}{\#\{\zeta^{i\kappa}\big\vert i=1,\ldots,\nu\}}.\) Hence the first assertion holds.

The second assertion is obtained by applying \(\pi\) to (3 ). ◻

**Proof. By arguing as in [2], we obtain that \(\eta(\mathcal{W})=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,\) where \[\begin{align} A=\frac{ \left| \begin{matrix} \partial_x(\delta_{12} h_3)y^{n_3}-\partial_y \delta_{12} & -\delta_{12}\\ \partial_x(\delta_{23} h_1)y^{n_1}-\partial_y \delta_{23} & -\delta_{23} \end{matrix} \right|}{\delta}, && B=\frac{ \left| \begin{matrix} \delta_{12}y^{n_3}h_3 & \partial_x(\delta_{12} h_3)y^{n_3}-\partial_y\delta_{12}\\ \delta_{23}y^{n_1}h_1 & \partial_x(\delta_{23} h_1)y^{n_1}-\partial_y\delta_{23} \end{matrix} \right| }{\delta}, \end{align}\] with \(\delta_{ij}=y^{n_j}h_j-y^{n_i}h_i\)  and  \(\delta = \delta_{12} \delta_{23} \delta_{31}.\)

When \(h_{12}(x,0)\not\equiv0\), we have \(y\not|h_{12}h_{23}h_{31}\); if \(n_1\leq n_2\leq n_3\), an argument from [2] shows that \[\begin{align} A(x,y)=\frac{n_2-n_1}{h_{31}(x,0)}\frac{1}{y^{n_1+1}}+\cdots, \qquad B(x,y)=\frac{n_1}{y}+\cdots. \end{align}\]

Assume now that \(h_{12}(x,0)\equiv0\). Then \(n_1=n_2\) and \(h_1(x,0)=h_2(x,0)\). Let us write \(h_2(x,y)=h_1(x,y)+y^ku(x,y)\) with \(k\geq1\) and \(u(x,0)\not\equiv0.\) Consider the case \(n_3\leq n_1=n_2\). A direct computation gives

It follows that \[\begin{align} A(x,y)=\frac{n_3-n_1-k}{h_{31}(x,0)}\frac{1}{y^{n_3+1}}+\cdots, \qquad B(x,y)=\left(n_3+k-\frac{kh_3(x,0)}{h_{31}(x,0)}\right)\frac{1}{y}+\cdots. \end{align}\]

If \(n_3<n_1=n_2\), then \(A(x,y)=\frac{n_3-n_1-k}{h_{3}(x,0)}\frac{1}{y^{n_3+1}}+\cdots\)  and  \(B(x,y)=\frac{n_3}{y}+\cdots.\)

If \(n_1=n_2=n_3\), we have by hypothesis \(h_3(x,0)=\lambda h_1(x,0)\) with \(\lambda\neq1\), hence \[\begin{align} A(x,y)=\frac{k}{(1-\lambda)h_1(x,0)}\frac{1}{y^{n_1+1}}+\cdots, \qquad B(x,y)=\left(n_1+\frac{k}{1-\lambda}\right)\frac{1}{y}+\cdots. \end{align}\]

The case \(n_3>n_1=n_2\) reduces to the case \(n_1=n_2=n_3\) with \(\lambda=0\), noting that \(\omega_3=\mathrm{d}x+y^{n_1}\tilde{h}_3(x,y)\mathrm{d}y\), where \(\tilde{h}_3(x,y)=y^{n_3-n_1}h_3(x,y).\) ◻

**Proof of Proposition 14. We will argue as in the proof of [2]. Consider again the local coordinate system \((U,(z,w))\) where \(C\cap U=\{w=0\}.\) We can then write \[\begin{align} \mathrm{T}\mathcal{W}_{d-\nu}|_{U}=\left\{\prod\limits_{l=1}^{d-\nu}(\mathrm{d}z+\tfrac{1}{\nu}g_{l}(z,w)\mathrm{d}w)=0\right\}. \end{align}\] First assume that \(\mathcal{W}_\nu\) is irreducible. By passing to the ramified covering \(\pi(x,y)=(z,w)=(x,y^{\nu})\) and using the second assertion of Lemma 13, we obtain that \(\pi^{*}\mathcal{W}_{\nu}=\boxtimes_{j=1}^{\nu}\mathcal{F}_j\) and \(\pi^{*}\mathcal{W}_{d-\nu}=\boxtimes_{l=1}^{d-\nu}\mathcal{G}_{l},\) where \[\begin{align} \mathcal{F}_j:\mathrm{d}x+\zeta^{-j\kappa}y^{\nu-\kappa-1}f(x,\zeta^jy)\mathrm{d}y=0,&& \mathcal{G}_{l}:\mathrm{d}x+y^{\nu-1}g_{l}(x,y^{\nu})\mathrm{d}y=0, \end{align}\] with \(1\leq\kappa<\nu\), \(\zeta=\exp(\tfrac{2\mathrm{i}\pi}{\nu})\) and \(f(x,0)\not\equiv0.\)

We then have \[\begin{align} K(\pi^*\mathcal{W})-K(\pi^*\mathcal{W}_\nu)-K(\pi^*\mathcal{W}_{d-\nu}) &=\sum_{\substack{1\le j<j'\le\nu\\ 1\le l\le d-\nu}}K(\mathcal{F}_j\boxtimes\mathcal{F}_{j'}\boxtimes\mathcal{G}_l)+\sum_{\substack{1\le j\le\nu\\ 1\le l<l'\le d-\nu}}K(\mathcal{F}_j\boxtimes\mathcal{G}_l\boxtimes\mathcal{G}_{l'})\\ &=\sum_{\substack{1\le j<j'\le\nu\\ 1\le l\le d-\nu}}\mathrm{d}\eta_{jj'l}+\sum_{\substack{1\le j\le\nu\\ 1\le l<l'\le d-\nu}}\mathrm{d}\eta_{jll'}. \end{align}\] where \(\eta_{jj'l}:=\eta(\mathcal{F}_j\boxtimes\mathcal{F}_{j'}\boxtimes\mathcal{G}_l)\)   and   \(\eta_{jll'}:=\eta(\mathcal{F}_j\boxtimes\mathcal{G}_l\boxtimes\mathcal{G}_{l'}).\)

Denoting by \(\varphi_{\ell}(x,y)=(x,\zeta^{\ell}y)\), \(\ell=1,\ldots,\nu,\) the Deck transformations of \(\pi\), we deduce that \[\begin{align} K(\pi^*\mathcal{W})-K(\pi^*\mathcal{W}_\nu)-K(\pi^*\mathcal{W}_{d-\nu}) &=\pi^*\Big(K(\mathcal{W})-K(\mathcal{W}_\nu)-K(\mathcal{W}_{d-\nu})\Big)\\ &=\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_\ell^*\pi^*\Big(K(\mathcal{W})-K(\mathcal{W}_\nu)-K(\mathcal{W}_{d-\nu})\Big)\\ &=\sum_{\substack{1\le j<j'\le\nu\\1\le l\le d-\nu}}\mathrm{d}\left(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_\ell^*\eta_{jj'l}\right) +\sum_{\substack{1\le j\le\nu\\ 1\le l<l'\le d-\nu}}\mathrm{d}\left(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_\ell^*\eta_{jll'}\right). \end{align}\] Note that if \(n\not\equiv0\mod\nu\) then \(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_{\ell}^*(y^{n}\mathrm{d}x)=0\), and if \(n\equiv -1 \mod \nu\) then \(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_{\ell}^*(y^{n}\mathrm{d}y)=y^{n}\mathrm{d}y\). Moreover, if we write \[\begin{align} \eta_{jj'l}=A_{jj'l}(x,y)\mathrm{d}x+B_{jj'l}(x,y)\mathrm{d}y &&\text{and}&& \eta_{jll'}=A_{jll'}(x,y)\mathrm{d}x+B_{jll'}(x,y)\mathrm{d}y, \end{align}\] Lemma 15 shows that each of \(A_{jj'l}\) and \(A_{jll'}\) has poles of order \(\leq\nu-\kappa\leq\nu-1\) along \(y=0\), and that each of \(B_{jj'l}\) and \(B_{jll'}\) is logarithmic along \(y=0\) with constant residue. It follows that \(\mathrm{d}\left(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_{\ell}^*\eta_{jj'l}\right)\) and \(\mathrm{d}\left(\frac{1}{\nu}\sum_{\ell=1}^{\nu}\varphi_{\ell}^*\eta_{jll'}\right)\) are holomorphic along \(\tilde{C}=\{y=0\}\), and hence so is \(K(\pi^*\mathcal{W})-K(\pi^*\mathcal{W}_\nu)-K(\pi^*\mathcal{W}_{d-\nu})\). Since \(\mathcal{W}_{d-\nu}\) is regular, \(K(\mathcal{W}_{d-\nu})\) is holomorphic along \(C\), and we deduce that \(K(\mathcal{W})-K(\mathcal{W}_{\nu})\) is also holomorphic along \(C.\)

Now assume that \(\mathcal{W}_\nu=\boxtimes_{\alpha=1}^{r}\mathcal{W}_{\nu_{\alpha}}\) with \(r\geq2\), each \(\mathcal{W}_{\nu_{\alpha}}\) being an irreducible \(\nu_{\alpha}\)-web having \(C\) as a weakly invariant curve. Consider the ramified covering \(\pi\colon(x,y)\mapsto(x,y^{\tilde{\nu}})\), where \(\tilde{\nu}:=\prod_{\alpha=1}^{r}\nu_{\alpha}.\) The pull-back of \(\mathcal{W}_{d-\nu}\) by \(\pi\) writes as \(\pi^{*}\mathcal{W}_{d-\nu}=\boxtimes_{l=1}^{d-\nu}\mathcal{G}_{l},\) where \[\mathcal{G}_{l}:\mathrm{d}x+y^{\tilde{\nu}-1}g_{l}(x,y^{\tilde{\nu}})\mathrm{d}y=0.\] As for \(\pi^{*}\mathcal{W}_{\nu}\), we have \(\pi=\pi_{\alpha}\circ\pi^{'}_{\alpha}\), where \(\pi_{\alpha}\colon(x,y)\mapsto(x,y^{\nu_{\alpha}})\) and \(\pi^{'}_{\alpha}\colon(x,y)\mapsto(x,y^{m_\alpha})\) with \(m_\alpha:= \frac{\tilde{\nu}}{\nu_\alpha}\). By Lemma 13, \(\pi_\alpha^*\mathcal{W}_{\nu_{\alpha}}=\boxtimes_{j=1}^{\nu_{\alpha}}\mathcal{F}_{j,0}^{\alpha}\), where \[\mathcal{F}_{j,0}^{\alpha}\,:\,\mathrm{d}x+\zeta_{\alpha}^{-j\kappa_{\alpha}}y^{\nu_{\alpha}-\kappa_{\alpha}-1}f_\alpha(x,\zeta_{\alpha}^{j}y)\mathrm{d}y=0,\] with \(1\leq\kappa_{\alpha}<\nu_\alpha\), \(\zeta_{\alpha}=\exp(\tfrac{2\mathrm{i}\pi}{\nu_{\alpha}})\) and \(f_\alpha(x,0)\not\equiv0\). It follows that \(\pi^*\mathcal{W}_{\nu}=\boxtimes_{\alpha=1}^{r}\boxtimes_{j=1}^{\nu_{\alpha}}\mathcal{F}_{j}^{\alpha},\) where \[\mathcal{F}_{j}^{\alpha}\,:\, \mathrm{d}x+m_{\alpha}\zeta_{\alpha}^{-j\kappa_{\alpha}}y^{\tilde{\nu}-\tilde{\kappa}_{\alpha}-1}f_{\alpha}(x,\zeta_{\alpha}^{j}y^{m_{\alpha}})\mathrm{d}y=0,\] with \(\tilde{\kappa}_{\alpha}:=m_{\alpha}\kappa_{\alpha}\),  \(1\leq\tilde{\kappa}_{\alpha}<\tilde{\nu}\).

We are thus reduced to a situation analogous to the case where \(\mathcal{W}_\nu\) was irreducible; the same argument shows that \(K(\mathcal{W})-K(\mathcal{W}_\nu)\) is holomorphic along \(C\). ◻

**Proof. We have \[\begin{align} K(\mathcal{W})-K(\mathcal{W}_n)=K(\mathcal{W}_{d-n}) +\sum_{\substack{1\le i<j\le n\\ 1\le k\le d-n}}K(\mathcal{F}_i\boxtimes\mathcal{F}_j\boxtimes\mathcal{F}^{'}_{k}) +\sum_{\substack{1\le i\le n\\ 1\le k<k'\le d-n}}K(\mathcal{F}_i\boxtimes\mathcal{F}^{'}_{k}\boxtimes\mathcal{F}^{'}_{k'}). \end{align}\] Now, \(K(\mathcal{F}_i\boxtimes\mathcal{F}_j\boxtimes\mathcal{F}^{'}_{k})\) and \(K(\mathcal{F}_i\boxtimes\mathcal{F}^{'}_{k}\boxtimes\mathcal{F}^{'}_{k'})\) are holomorphic along \(C\) by applying [2]. The lemma then follows from the assumption that \(K(\mathcal{W}_{d-n})\) is holomorphic along \(C.\) ◻

**Proof of Theorem 6. In a neighborhood of a generic point of \(C\), we can decompose \(\mathcal{W}_n\) as \(\mathcal{W}_n=\mathcal{W}_{\nu}\boxtimes\mathcal{W}_{\nu'}\), with \(\mathcal{W}_{\nu}=\boxtimes_{\alpha=1}^{r}\mathcal{W}_{\nu_{\alpha}}\) and \(\mathcal{W}_{\nu'}=\boxtimes_{\beta=1}^{s}\mathcal{W}_{\nu^{'}_{\beta}}\), where each \(\mathcal{W}_{\nu_{\alpha}}\) (resp. \(\mathcal{W}_{\nu^{'}_{\beta}}\)) is an irreducible \(\nu_{\alpha}\)-web (resp. \(\nu^{'}_{\beta}\)-web) admitting \(C\) as a strongly (resp. weakly) invariant curve. By choosing local coordinates \((z,w)\) such that \(C=\{w=0\}\) and passing to the ramified covering \(\pi\colon(x,y)\mapsto(z,w)=(x,y^{\tilde{\nu}})\), where \(\tilde{\nu}=\displaystyle\prod_{\alpha=1}^{r}\nu_{\alpha}\displaystyle\prod_{\beta=1}^{s}\nu^{'}_{\beta}\), we obtain that \(\pi^*\mathcal{W}_\nu=\boxtimes_{i=1}^{\nu}\mathcal{F}_i\), \(\pi^*\mathcal{W}_{\nu'}=\boxtimes_{j=1}^{\nu'}\mathcal{G}_j\) and \(\pi^*\mathcal{W}_{d-n}=\boxtimes_{k=1}^{d-n}\mathcal{H}_{k}\), where the \(\mathcal{F}_i\) are foliations having \(\tilde{C}=\{y=0\}\) as an invariant curve, and the \(\mathcal{G}_{j}\) and \(\mathcal{H}_{k}\) are foliations transverse to \(\tilde{C}.\)

We then have

Note that, by [2], the curvatures \(K(\mathcal{F}_i \boxtimes\mathcal{F}_{i'}\boxtimes \mathcal{H}_k)\), \(K(\mathcal{F}_i\boxtimes\mathcal{G}_j\boxtimes\mathcal{H}_k)\) and \(K(\mathcal{F}_i \boxtimes\mathcal{H}_k\boxtimes\mathcal{H}_{k'})\) are holomorphic along \(\tilde{C}\). Moreover, \(K\big(\pi^*(\mathcal{W}_{\nu'}\boxtimes\mathcal{W}_{d-n})\big)-K(\pi^*\mathcal{W}_{\nu'})=\pi^*\Big(K(\mathcal{W}_{\nu'}\boxtimes\mathcal{W}_{d-n})-K(\mathcal{W}_{\nu'})\Big)\) is holomorphic on \(\tilde{C}\), thanks to Proposition 14. It follows that \(K(\pi^*\mathcal{W})-K(\pi^*\mathcal{W}_n)\) is holomorphic along \(\tilde{C}\), and therefore \(K(\mathcal{W})-K(\mathcal{W}_n)\) is holomorphic along \(C.\) ◻

**Proof of Proposition 7. We argue as in the proof of Proposition 14, using the same notation. Writing \(\mathcal{W}_\nu=\boxtimes_{\alpha=1}^{r}\mathcal{W}_{\nu_\alpha}\) and passing to the ramified covering \(\pi\colon(x,y)\mapsto(z,w)=(x,y^{\tilde{\nu}})\), where \(\tilde{\nu}=\prod_{\alpha=1}^{r}\nu_\alpha\), we obtain that \(\pi^*\mathcal{W}_{\nu}=\boxtimes_{\alpha=1}^{r}\boxtimes_{j=1}^{\nu_{\alpha}}\mathcal{F}_{j}^{\alpha},\) where each \(\mathcal{F}_j^\alpha\) is a foliation transverse to \(\tilde{C}=\{y=0\}\) and defined by \[\omega_j^\alpha=\mathrm{d}x+m_{\alpha}\zeta_{\alpha}^{-j\kappa_{\alpha}}y^{n_\alpha}f_{\alpha}(x,\zeta_{\alpha}^{j}y^{m_{\alpha}})\mathrm{d}y,\] with \(n_\alpha=\tilde{\nu}-\tilde{\kappa}_\alpha-1\), \(\tilde{\kappa}_\alpha=m_\alpha\kappa_\alpha=\tilde{\nu}\rho_\alpha\), \(\rho_\alpha=\frac{\kappa_\alpha}{\nu_\alpha}<1\), \(1\le\tilde{\kappa}_\alpha<\tilde{\nu}\)  and  \(f_\alpha(x,0)\not\equiv0.\)

Let us denote \(\eta^{\alpha\beta\gamma}_{jj'j''}=\eta(\mathcal{F}_j^\alpha\boxtimes\mathcal{F}_{j'}^\beta\boxtimes\mathcal{F}_{j''}^\gamma)\); we can write \[K(\pi^*\mathcal{W}_\nu)=\sum_{1\le\alpha\le\beta\le\gamma\le r} \sum_{\substack{1\le j\le\nu_\alpha\\ 1\le j'\le\nu_\beta\\ 1\le j''\le\nu_\gamma}} \mathrm{d} \eta^{\alpha\beta\gamma}_{jj'j''}.\]

Setting \(\varphi_{\ell}(x,y)=(x,\zeta^{\ell}y)\), \(\ell=1,\ldots,\tilde{\nu},\) where \(\zeta=\exp(\tfrac{2\mathrm{i}\pi}{\tilde{\nu}}),\) we have \[\begin{align} K(\pi^*\mathcal{W}_\nu) =\pi^*K(\mathcal{W}_\nu) =\frac{1}{\tilde{\nu}}\sum_{\ell=1}^{\tilde{\nu}}\varphi_\ell^*\pi^*K(\mathcal{W}_\nu) =\sum_{1\le\alpha\le\beta\le\gamma\le r}\sum_{\substack{1\le j\le\nu_\alpha\\1\le j'\le\nu_\beta\\1\le j''\le \nu_\gamma}}\mathrm{d} \left( \frac{1}{\tilde{\nu}}\sum_{\ell=1}^{\tilde{\nu}}\varphi_\ell^*\eta^{\alpha\beta\gamma}_{jj'j''} \right). \end{align}\]

Using conditions (\(\mathfrak{a}\)), resp. (\(\mathfrak{b}\)), resp. (\(\mathfrak{c}\)), we easily verify that the \(3\)-webs \(\mathcal{F}_j^\alpha\boxtimes\mathcal{F}_{j'}^\alpha\boxtimes\mathcal{F}_{j''}^\alpha\), resp. \(\mathcal{F}_j^\alpha\boxtimes\mathcal{F}_{j'}^\alpha\boxtimes\mathcal{F}_{j''}^\gamma\) with \(\alpha\neq\gamma\), resp. \(\mathcal{F}_j^\alpha\boxtimes\mathcal{F}_{j'}^\beta\boxtimes\mathcal{F}_{j''}^\gamma\) with \(\alpha<\beta<\gamma\), have the form of Lemma 15, so that \[\eta_{jj'j''}^{\alpha\beta\gamma}=\left(\frac{g_{jj'j''}^{\alpha\beta\gamma}(x)}{y^{n_{\alpha\beta\gamma}+1}}+\cdots\right)\mathrm{d}x +\left(\frac{c_{jj'j''}^{\alpha\beta\gamma}}{y}+\cdots\right)\mathrm{d}y,\] where \(n_{\alpha\beta\gamma}=\min(n_\alpha,n_\beta,n_\gamma)\), \(g_{jj'j''}^{\alpha\beta\gamma}\) is a holomorphic function of \(x\) and \(c_{jj'j''}^{\alpha\beta\gamma}\in\mathbb{C}.\) Since \(n_{\alpha\beta\gamma}+1\le\tilde{\nu}-1\), we deduce (cf. proof of Proposition 14) that \(\mathrm{d}\Big(\frac{1}{\tilde{\nu}}\sum_{\ell=1}^{\tilde{\nu}}\varphi_\ell^*\eta_{jj'j''}^{\alpha\beta\gamma}\Big)\) is holomorphic along \(\tilde{C}=\{y=0\}\), and hence so is \(K(\pi^*\mathcal{W}_\nu)\). Consequently, \(K(\mathcal{W}_\nu)\) is holomorphic along \(C=\{w=0\}\). ◻

**Proof of Corollary 9. If \(\nu=2\), [2] ensures that the curvature of \(\mathcal{W}\) is holomorphic along \(C\).

Assume that \(\nu\geq3.\) According to the first assertion of Lemma 13, we have \[\kappa(\nu-1)\leq\mathrm{mult}\left(\Delta(\mathcal{W}_{\nu}),C\right)<3(\nu-1),\] hence \(\kappa<3\), and therefore \(\kappa\in\{1,2\}\subset\{1,\ldots,\nu-1\}\). It follows that \(C\) is weakly invariant by \(\mathcal{W}_\nu\) and that \(\gcd(\nu,\kappa)\leq2.\) The web \(\mathcal{W}_\nu\) is then non-degenerate along \(C.\) By Proposition 7, \(K(\mathcal{W}_\nu)\) is holomorphic on \(C\), and by Theorem 6, the same holds for \(K(\mathcal{W})\). ◻

**Proof of Corollary 10. By Lemma 13, we have \[\kappa(\nu-1)\leq\mathrm{mult}\left(\Delta(\mathcal{W}_{\nu}),C\right)<\nu(\nu-1),\] hence \(\kappa<\nu\). The curve \(C\) is then weakly invariant by \(\mathcal{W}_\nu.\) Furthermore, the assumption on \(\nu\) and the inequality \(\kappa<\nu\) imply that \(\gcd(\nu,\kappa)\leq2.\) It follows that \(\mathcal{W}_\nu\) is non-degenerate along \(C\), so that \(K(\mathcal{W}_\nu)\) is holomorphic along \(C\) by Proposition 7, and so is \(K(\mathcal{W})\) by Theorem 6. ◻

**Proof of Theorem 11. Let us decompose \(\mathcal{W}_{n}^{\rm{str}}\) and \(\mathcal{W}_{n}^{\rm{wk}}\) into \(\mathcal{W}_{n}^{\rm{str}}=\boxtimes_{\alpha=1}^{r}\mathcal{W}_{\nu_\alpha}\) and \(\mathcal{W}_{n}^{\rm{wk}}=\boxtimes_{\beta=1}^{s}\mathcal{W}_{\nu^{'}_{\beta}}\), where each \(\mathcal{W}_{\nu_{\alpha}}\) (resp. \(\mathcal{W}_{\nu^{'}_{\beta}}\)) is an irreducible \(\nu_{\alpha}\)-web (resp. \(\nu^{'}_{\beta}\)-web) having \(C\) as a strongly (resp. weakly) invariant curve. We take local coordinates \((z,w)\) such that \(C=\{w=0\}\). By passing to the ramified covering \(\pi\colon(x,y)\mapsto(z,w)=(x,y^{\nu})\), where \(\nu=\displaystyle\prod_{\alpha=1}^{r}\nu_{\alpha}\displaystyle\prod_{\beta=1}^{s}\nu^{'}_{\beta}\), we obtain that \(\pi^*\mathcal{W}_{n}^{\rm{str}}=\boxtimes_{i=1}^{n_1}\mathcal{F}_i\) and \(\pi^*(\mathcal{W}_{n}^{\rm{wk}}\boxtimes\mathcal{W}_{d-n})=\boxtimes_{j=1}^{d-n_1}\mathcal{F}'_j\), where the \(\mathcal{F}_i\) are foliations having \(\tilde{C}=\{y=0\}\) as an invariant curve, the \(\mathcal{F}'_{j}\) are foliations transverse to \(\tilde{C}\), and \(n_1=\sum_{\alpha=1}^{r}\nu_\alpha.\)

The web \(\mathcal{W}_{n}^{\rm{wk}}\) being non-degenerate along \(C\), Proposition 7 ensures that \(K(\mathcal{W}_{n}^{\rm{wk}})\) is holomorphic on \(C\). Since \(\mathcal{W}_{d-n}\) is regular and transverse to \(C\), Theorem 6 implies that \(K(\mathcal{W}_{n}^{\rm{wk}}\boxtimes\mathcal{W}_{d-n})\) is also holomorphic on \(C.\) It follows that \(K\big(\pi^*(\mathcal{W}_{n}^{\rm{wk}}\boxtimes\mathcal{W}_{d-n})\big)\) is holomorphic along \(\tilde{C}.\) We can then apply Lemma 16 to the web \(\pi^*\mathcal{W}=\pi^*\mathcal{W}_{n}^{\rm{str}}\boxtimes\pi^*(\mathcal{W}_{n}^{\rm{wk}}\boxtimes\mathcal{W}_{d-n})\) and deduce that \(K(\pi^*\mathcal{W})-K(\pi^*\mathcal{W}_{n}^{\rm{str}})\) is holomorphic on \(\tilde{C}.\) Consequently, \(K(\mathcal{W})-K(\mathcal{W}_{n}^{\rm{str}})\) is holomorphic along \(C\). ◻