Effective Integrability of Lins Neto's Family of Foliations

TL;DR

This paper provides an algorithmic approach to analyze the degree, singularities, and generators of elliptic curves in a family of complex foliations, utilizing computational tools and Cremona transformations.

Contribution

It introduces a method to explicitly compute properties of elliptic curves in Lins Neto's family of foliations for any parameter t, combining algebraic geometry and computational software.

Findings

Algorithmically determines degree, singularities, and multiplicities of elliptic curves.

Provides explicit formulas for generators of elliptic pencils.

Uses Cremona maps to analyze foliation transformations.

Abstract

A. Lins Neto presented in [Lins-Neto,2002] a $1$-dimensional family of degree four foliations on the complex projective plane $\mathcal{F}_{t \in \overline{\mathbb{C}}}$ with non-degenerate singularities of fixed analytic type, whose set of parameters $t$ for which $\mathcal{F}_t$ is an elliptic pencil is dense and countable. In [McQuillan,2001] and [Guillot,2002], M. McQuillan and A. Guillot showed that the family lifts to linear foliations on the abelian surface $E \times E$, where $E = \mathbb{C}/\Gamma$, $\Gamma = < 1 , \tau>$ and $\tau$ is a primitive 3rd root of unity, the parameters for which $\mathcal{F}_t$ are elliptic pencils being $t\in \mathbb{Q}(\tau) \cup {\infty}$. In [Puchuri,2013], the second author gave a closed formula for the degree of the elliptic curves of $\mathcal{F}_t$ a function of $t \in \mathbb{Q}(\tau)$. In this work we determine degree, positions and multiplicities of singularities of the elliptic curves of $\mathcal{F}_t$, for any given $t \in \mathbb{Z}(\tau)$ in algorithmical way implemented in Python. And also we obtain the explicit expressions for the generators of the elliptic pencils, using the Singular software. Our constructions depend on the effect of quadratic Cremona maps on the family of foliations $\mathcal{F}_t$.