arXiv subjects

Beatriz Molina-Samper

Publications and source records attributed to Beatriz Molina-Samper.

6 recordsLinked to original sources

arXiv2025

Radial foliations in dimension three

Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C3, 0),E,F) is almost radial when it has a reduction of singularities without singular points; it will be "radial" under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the "open book" situation. We end the paper with a discussion on the general almost radial case.

math.AG↗

arXiv2023

Idealistic Flowers in the Reduction of Singularities

We present here a proof of the classical reduction of singularities based on the idea of "idealistic flowers". We follow the general ideas of Maximal Contact Theory, presented in a recent book of Aroca, Hironaka and Vicente, that recovers three old publications of Jorge Juan Institute. The concept of idealistic flowers deals with the globalization problems arising from the local nature of the maximal contact.

math.AG↗

arXiv2020

Invariant Surfaces for Toric Type Foliations in Dimension Three

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0), without saddle-nodes, has invariant surface. We extend the argument of Cano-Cerveau, done for the nondicritical case, to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to global curves. We build the invariant surface as a germ along the singular locus and those global invariant curves. The result of Ortiz-Rosales-Voronin, about the distribution of invariant curves in dimension two, is a key argument in our proof.

math.AG↗

arXiv2020

Newton Non-degenerate Foliations and Blowing-ups

A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.

math.DS↗

arXiv2019

Global Invariant Branches of Non-degenerate Foliations on Projective Toric Surfaces

We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type foliations in terms of "non-degeneracy" conditions, associated to Newton polygons. We also give a description of the relationship between invariant algebraic curves and isolated invariant branches, valid for the case of toric type, by means of the following dichotomy. Either there is a rational first integral and there are no isolated invariant branches or we have only finitely many global invariant curves, all of them extending isolated invariant branches.

math.AG↗

arXiv2018

Combinatorial Aspects of Classical Resolution of Singularities

We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.

math.AG↗