Algebraic curves and foliations

TL;DR

This paper studies the structure of holomorphic foliations invariant under algebraic curves with quasi-homogeneous singularities, proving they are generated by at most four elements and often by two over suitable field extensions, linking to free divisors.

Contribution

It provides an explicit, algorithmic construction of generators for the module of foliations fixing a given algebraic curve with quasi-homogeneous singularities, and shows these modules are often free over field extensions.

Findings

The module of foliations fixing the curve is generated by at most four elements.

Over a suitable field extension, the module is generated by two elements.

In many examples, the base field suffices for generation.

Abstract

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${mathbb A}^2_\k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension $K$ of $k$ such a module over $K[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, $K=k$.