Algebraic integrability of foliations of the plane (Q858696)

Let \({\mathbb K}\) be an algebraically closed field of characteristic \(0\). A foliation \({\mathcal F}\) of \({\mathbb P}^2({\mathbb K})\) is the datum, for some \(d\geq 0\), of a global section \(\Omega\) with only isolated zeros of the sheaf of \(\Omega^1(d)_{{\mathbb P}^2({\mathbb K})}\), up to multiplication by a nonzero scalar. A rational first integral of the foliation \({\mathcal F}\) is a rational function \(R\) on the plane such that \(dR\wedge \Omega\equiv 0\). Given a foliation \({\mathcal F}\) of the plane, there is a so-called minimal resolution of \({\mathcal F}\), namely a blow up \(X_0\) of the plane such that the strict transform of \({\mathcal F}\) on \(X_0\) has only a certain type of singularities (``simple singularities'') which cannot be improved by further blowing up and such that \(X_0\) is minimal with respect to this property. The points blown up to get \(X_0\) are of two types: dicritical and non dicritical (we refer the reader to the paper for the definition). We denote by \(X\) the blow up of the plane at the dicritical singular points of \({\mathcal F}\). Under the assumption that the Mori cone of \(X\) is polyhedral, the authors give an algorithm for deciding whether a rational first integral of \({\mathcal F}\) exists and for computing it in case it exists. The assumption that the Mori cone is polyhedral is satisfied whenever \(X\) is the blow up of \({\mathbb P}^2({\mathbb K})\) at at most 8 points. More generally, it is possible to decide whether the assumption holds by looking at the proximity graph associated with the blow up \(X\to {\mathbb P}^2({\mathbb K})\).