Session S39 - Differential Equations and Geometric Structures

Wednesday, July 14, 12:00 ~ 12:50 UTC-3

Poincaré Problem for foliations on $\mathbb{CP}^2$ with a unique singularity

Claudia Reynoso Alcántara

Universidad de Guanajuato, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Poincaré problem for foliations on $\mathbb{CP}^2$ is a very famous problem that asks the following: if we have a foliation on $\mathbb{CP}^2$ of degree $d$ with a rational first integral of degree $s$ then, can we bound $s$ as function of $d$? The answer in general is that it is not possible. Consider for example the foliation on $\mathbb{CP}^2$ of degree 1 given by the 1-form:

\begin{equation*} pyzdx+qxzdy-(p+q)yxdz \end{equation*}

with $p$ and $q$ positive integers, the pencil $\{\alpha x^py^q - \beta z^{p+q}\}$ of degree $p+q$ defines the rational first integral.

The main objective of this talk is to give a positive answer to this problem for the case of foliations on $\mathbb{CP}^2$ of degree $d$ with a unique singular point. To obtain the bound we use the geometry of the local singular scheme of the foliation. We will give examples of extreme cases: when de degree of the rational first integral is the maximum and minimum possible.

Joint work with Ramón Ronzón-Lavié (University of Toronto).

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