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

Hal Schenck (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Hiroaki Terao (Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan)

Masahiko Yoshinaga (Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan)

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

Published 25 March 2019