Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on ℙ2ℂ
Abstract
Let 𝒲=𝒲nboxtimes𝒲d−n be a d-web on (ℂ2,0), where 𝒲n is an n-web admitting an irreducible totally invariant~curve~C, and 𝒲d−n is a regular (d−n)-web transverse to C. We show that the curvature of 𝒲 is holomorphic along~C if and only if the curvature of 𝒲n is. For each irreducible subweb 𝒲ν of 𝒲n, we distinguish two cases depending on whether C is strongly or weakly invariant by 𝒲ν; in the latter case, we introduce the \textsc{Puiseux} index and exponent i(𝒲ν,C) and ρ(𝒲ν,C). These invariants allow us to define the notion of non-degeneracy of 𝒲n along C. When 𝒲n is non-degenerate along C, we prove that K(𝒲n), and hence K(𝒲), is holomorphic along C, generalizing a result of \textsc{Mar\'{\i}n} and~\textsc{Pereira}. Moreover, we obtain a natural decomposition of 𝒲n as the product of two subwebs 𝒲n=𝒲nstrboxtimes𝒲nwk. Under the assumption that 𝒲nwk is non-degenerate along C, we show that the holomorphy of K(𝒲) on C is equivalent to that of K(𝒲nstr). As an application, we establish a flatness result for products of dual webs of reduced convex foliations on ℙ2ℂ.
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BibTeX
@misc{bedrouni2026holomorphy,
title = {{Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on $\mathbbP^2_\mathbbC$}},
author = {Samir Bedrouni},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.13373},
eprint = {2606.13373},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05621001v3},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on ℙ^2_ℂ
AU - Samir Bedrouni
PY - 2026
DA - 2026/09/14
PB - HAL
DO - 10.48550/arxiv.2606.13373
UR - https://hal.science/hal-05621001v3
AB - Let 𝒲=𝒲_nboxtimes𝒲_d−n be a d-web on (ℂ^2,0), where 𝒲_n is an n-web admitting an irreducible totally invariant~curve~C, and 𝒲_d−n is a regular (d−n)-web transverse to C. We show that the curvature of 𝒲 is holomorphic along~C if and only if the curvature of 𝒲_n is. For each irreducible subweb 𝒲_ν of 𝒲_n, we distinguish two cases depending on whether C is strongly or weakly invariant by 𝒲_ν; in the latter case, we introduce the \textsc{Puiseux} index and exponent i(𝒲_ν,C) and ρ(𝒲_ν,C). These invariants allow us to define the notion of non-degeneracy of 𝒲_n along C. When 𝒲_n is non-degenerate along C, we prove that K(𝒲_n), and hence K(𝒲), is holomorphic along C, generalizing a result of \textsc{Mar\'{\i}n} and~\textsc{Pereira}. Moreover, we obtain a natural decomposition of 𝒲_n as the product of two subwebs 𝒲_n=𝒲_n^strboxtimes𝒲_n^wk. Under the assumption that 𝒲_n^wk is non-degenerate along C, we show that the holomorphy of K(𝒲) on C is equivalent to that of K(𝒲_n^str). As an application, we establish a flatness result for products of dual webs of reduced convex foliations on ℙ^2_ℂ.
ER -