arXiv subjects

Wodson Mendson

Publications and source records attributed to Wodson Mendson.

8 recordsLinked to original sources

arXiv2026

Explicit integration, birational models and symmetries of Lins Neto's exceptional families of foliations

In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincar\'e Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.

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arXiv2026

Foliations with small singular set in arbitrary characteristic

This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic $p \ge 0$. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, such foliations are $p$-closed. However, there are regular foliations on weak del Pezzo surfaces, in characteristic 2, which are not $p$-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be $p$-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for $p$-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.

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arXiv2025

Non-algebraicity of foliations via reduction modulo $2$

Motivated by the Jouanolou foliation problem, we investigate the non-algebraicity of foliations by curves on $\mathbb{P}^2_{\mathbb{C}}$. We present a criterion to show that such a foliation has no algebraic invariant curves, using a method of reduction modulo $2$. Finally, using this criterion, we give a new proof that the Jouanolou foliation of odd degree has no algebraic invariant curves. We also present other classes of foliations without algebraic invariant curves.

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arXiv2025

The space of foliations on projective spaces in positive characteristic

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces varies with the characteristic of the base field in very low degrees and establishes an arbitrary characteristic version of Calvo-Andrade's stability of generic logarithmic $1$-forms under deformation.

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arXiv2023

Arithmetic aspects of the Jouanolou foliation

We investigate the structure of the $p$-divisor for the Jouanolou foliation where we show, under some conditions, that it can be irreducible or has a $p$-factor. We study the reduction modulo $p$ of foliations on the projective plane and its applications to the problems of holomorphic foliations. We give new proof, via reduction modulo $2$, of the fact that the Jouanolou foliation on the complex projective plane of odd degree, under some arithmetic conditions, has no algebraic solutions.

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arXiv2022

Codimension one foliations in positive characteristic

We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.

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arXiv2022

Foliations on smooth algebraic surfaces over positive characteristic

We investigate the notion of the $p$-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic $p$ and we study some of their properties. We present a structure theorem for the $p$-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such $p$-divisors are reduced.

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arXiv2018

Some arithmetic aspects of polynomial maps

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ ($n\geq2$) with jacobian $1$ is an automorphism. We present a survey about some results around this conjecture and we discuss an arithmetic aspect of this conjecture due to Essen-Lipton. We investigate some cases of this arithmetic approach showing the close relationship between the Jacobian Conjecture and the problem of counting $\mathbb{F}_p$-points of an affine scheme.

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