On the classification of foliations of degree three with one singularity
A degree-\(d\) foliation on \(\mathbb{P}^2\) can be defined by a homogeneous vector field \(v\) of degree \(d\), not vanishing in codimension one. It is known that \(v\) must vanish at \(d^2 + d+ 1\) points, called singularities, counted with multiplicities; for a generic \(v\) they are all distinct. Foliations with only one singularity (of multiplicity \(d^2 + d+ 1\)) are rather special and their classification is an interesting problem, of which the article under review solves a particular case. Precisely, the author proves that any foliation of degree \(d=3\) with nontrivial finite isotropy group and only one singularity must fit, up to change of coordinates, in one of 7 families of foliations \(\mathcal{F}_j\), \(1\leq j \leq 7\) described in the paper. The proof is based on an algorithm written for AXIOM.
scientific article; zbMATH DE number 7493809