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

Abstract

Let $\mathcal{W}={\mathcal{W}}_n⊠{\mathcal{W}}_{d-n}$ 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\'ın} 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 ${\mathcal{W}}_n={\mathcal{W}}_n^{\mathrm{str}}⊠{\mathcal{W}}_n^{\mathrm{wk}}.$ Under the assumption that ${\mathcal{W}}_n^{\mathrm{wk}}$ is non-degenerate along $C$, we show that the holomorphy of $K(\mathcal{W})$ on $C$ is equivalent to that of $K({\mathcal{W}}_n^{\mathrm{str}}).$

12 pages. A first version of this paper appeared on HAL: https://hal.science/hal-05621001v1