1. A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$.
- Author
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Bedrouni, Samir and Marín, David
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WEB-based user interfaces , *CURVATURE , *INTEGERS - Abstract
Let d≥3$d\ge 3$ be an integer. For a holomorphic d$d$‐web W$\mathcal {W}$ on a complex surface M$M$, smooth along an irreducible component D$D$ of its discriminant Δ(W)$\Delta (\mathcal {W})$, we establish an effective criterion for the holomorphy of the curvature of W$\mathcal {W}$ along D$D$, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) LegH$\mathrm{Leg}\mathcal {H}$ of a homogeneous foliation H$\mathcal {H}$ of degree d$d$ on PC2$\mathbb {P}^{2}_{\mathbb {C}}$, generalizing some of our previous results. This then allows us to study the flatness of the d$d$‐web LegH$\mathrm{Leg}\mathcal {H}$ in the particular case where the foliation H$\mathcal {H}$ is Galois. When the Galois group of H$\mathcal {H}$ is cyclic, we show that LegH$\mathrm{Leg}\mathcal {H}$ is flat if and only if H$\mathcal {H}$ is given, up to linear conjugation, by one of the two 1‐forms ω1d=yddx−xddy$\omega _1^{d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, ω2d=xddx−yddy$\omega _2^{d}=x^d\mathrm{d}x-y^d\mathrm{d}y$. When the Galois group of H$\mathcal {H}$ is noncyclic, we obtain that LegH$\mathrm{Leg}\mathcal {H}$ is always flat. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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