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# Mathematical Research Letters

## Volume 25 (2018)

### Number 6

### Logarithmic vector fields for curve configurations in $\mathbb{P}^2$ with quasihomogeneous singularities

Pages: 1977 – 1992

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n6.a14

#### Authors

#### Abstract

Let $\mathcal{A} = \bigcup^{r}_{i=1} C_i \subseteq \mathbb{P}^2_C$ be a collection of plane curves, such that each singular point of $\mathcal{A}$ is quasihomogeneous. We prove that if $C$ is an irreducible curve having only quasihomogeneous singulartities, such that $C \cap \mathcal{A} \subseteq C_{sm}$ and every singular point of $\mathcal{A}\cup C$ is quasihomogeneous, then there is a short exact sequence relating the $\mathcal{O}_{\mathbb{P}^2}$-module $\mathrm{Der} (-\log \mathcal{A})$ of vector fields on $\mathbb{P}^2$ tangent to $\mathcal{A}$ to the module $\mathrm{Der}(-\log \mathcal{A} \cup C)$. This yields an inductive tool for studying the splitting of the bundles $\mathrm{Der}(-\log \mathcal{A})$ and $\mathrm{Der}(-\log \mathcal{A} \cup C)$, depending on the geometry of the divisor $\mathcal{A}\vert {}_C$ on $C$.

Received 18 November 2013

Accepted 14 August 2015

Published 25 March 2019