Algebraic integrability of foliations by extension to Hirzebruch surfaces. Applications to bounded negativity.

Abstract

We make progress on two open mathematical problems: the problem of algebraic integrability of polynomial foliations on $\mathbb{C}^2$ and the bounded negativity conjecture. For the first one, we identify $\mathbb{C}^2$ with an open set of $\mathbb{P}^2$ or $\mathbb{F}_{\delta}$, $\delta\geq0$, and study foliations $\mathcal{F}$ on these surfaces whose local form is isomorphic to the affine foliation. We obtain necessary conditions for algebraic integrability by studying the sky of the dicritical configuration of $\mathcal{F}$. We propose algorithms that solve the first problem under some conditions. For the second one, we consider a rational surface $S$ and an integral curve $H$ on $S$. If $S$ is obtained from $\mathbb{F}_{\delta}$ (respectively, $\mathbb{P}^2$), we provide a bound on $\frac{H^2}{H\cdot (F^*+M^*)}$ (respectively, on $\frac{H^2}{(H\cdot L^*)^2}$ and on $\frac{H^2}{H\cdot L^*}$), where $F^*$, $M^*$ and $L^*$ are the total transforms of a general fiber, a section of self-intersection $\delta$ of $\mathbb{F}{\delta}$ and a general line of $\mathbb{P}^2$ respectively.