arXiv subjects

Ariel Molinuevo

Publications and source records attributed to Ariel Molinuevo.

11 recordsLinked to original sources

arXiv2023

Rational pullbacks of toric foliations

This article is dedicated to the study of singular codimension $1$ foliations $\mathcal{F}$ on a simplicial complete toric variety $X$ and their pullbacks by dominant rational maps $φ:\mathbb{P}^n\dashrightarrow X$. First, we describe the singularities of $\mathcal{F}$ and $φ^*\mathcal{F}$ for a generic pair $(φ,\mathcal{F})$. Then we show that the first order deformations of $φ^*\mathcal{F}$ arising from first order unfoldings are the families of the form $φ_\varepsilon^*\mathcal{F}$, where $φ_\varepsilon$ is a perturbation of $φ$. We also prove that the deformations of the form $φ^*\mathcal{F}_\varepsilon$ consist exactly of the families which are tangent to the fibers of $φ$. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.

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arXiv2022

Stability of pullbacks of foliations on weighted projective spaces

We show a stability-type theorem for foliations on projective spaces which arise as pullbacks of foliations with a split tangent sheaf on weighted projective spaces. As a consequence, we will be able to construct many irreducible components of the corresponding spaces of foliations, most of them being previously unknown. This result also provides an alternative and unified proof for the stability of other families of foliations.

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arXiv2022

Singular locus of q-logarithmic foliations

We determine the structure of the singular locus of generic codimension-$q$ logarithmic foliations and its relation with the unfoldings of said foliations. In the case where the ambient variety is the projective space $\mathbb{P}^n$ we calculate the graded ideal defining the scheme of persistent singularities.

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arXiv2021

Families and unfoldings of singular holomorphic Lie Algebroids

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.

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arXiv2021

Foliations with persistent singularities

Let $ω$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $ω$ in various settings. We relate a certain type of singularities, which we name \emph{persistent}, with the unfoldings of $ω$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making computations on the Cox ring and modules over this ring.

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arXiv2018

On the Camacho-Lins Neto regularity

We work with codimension one foliations in the projective space $\mathbb{P}^{n}$, given a differential one form $ω\in H^0(\mathbb{P}^n,Ω^1_{\mathbb{P}^n}(e))$, such differential form verifies the Frobenius integrability condition $ω\wedge dω=0$. In this work we show that the Camacho-Lins Neto regularity, applied for $ω$, is equivalent to the fact that every first order unfolding of $ω$ is trivial up to isomorphism. We do this by computing the Castelnuovo-Mumford regularity of the ideal $I(ω)$ of first order unfoldings. With this result, we are also showing that the only regular projective foliations, with reduced singular locus, are the ones that have singular locus only Kupka type singularities. At last we use these results to show that every foliation $\varpi\in Ω^1_{\mathbb{C}^{n+1}}$, with initial form $ω$ regular and dicritical, is isomorphic to $ω$.

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arXiv2018

On first order deformations of homogeneous foliations

We study analytic deformations of holomorphic foliations given by homogeneous integrable one-forms in the complex affine space $\mathbb C^n$. The deformation is supposed to be of first order (order one in the parameter). We also assume that the deformation is given by homogeneous polynomial one-forms. The deformation takes place in the affine space since we are not assuming that the foliations descent to the projective space. We describe the space of such deformations in three main situations: (1) the given foliation is given by the level hypersurfaces of a homogeneous polynomial. (2) the foliation is rational, ie., has a first integral of type $P^r/Q^s$ for some homogeneous polynomials $P,Q$. (3) the foliation is logarithmic of a generic type. We prove that, for each class above, the first order homogeneous deformations of same degree are in the very same class. We also investigate the existence of such deformations with different degree.

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arXiv2018

On the geometry of the singular locus of a codimension one foliation in $\mathbb{P}^n$

We will work with codimension one holomorphic foliations over the complex projective space, represented by integrable forms $ω\in H^0(Ω^1_{\PP^n}(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $ω\in H^0(Ω^1_{\PP^3}(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen-Macaulay. As a consequence, we prove the connectedness of the Kupka set in $\PP^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.

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arXiv2017

The Kupka Scheme and Unfoldings

Let $ω$ be a differential 1-form defining an algebraic foliation of codimension 1 in projective space. In this article we use commutative algebra to study the singular locus of $ω$ through its ideal of definition. Then, we expose the relation between the ideal defining the Kupka components of the singular set of $ω$ and the first order unfoldings of $ω$. Exploiting this relation, we show that the set of Kupka points of $ω$ is generically not empty. As an application of this results, we can compute the ideal of first order unfoldings for some known components of the space of foliations.

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arXiv2016

Deformations of the Exterior Algebra of Differential Forms

Let $D:Ω\xrightarrow{}Ω$ be a differential operator defined in the exterior algebra $Ω$ of differential forms over the polynomial ring $S$ in $n$ variables. In this work we give conditions for deforming the module structure of $Ω$ over $S$ induced by the differential operator $D$, in order to make $D$ an $S$-linear morphism while leaving the $\mathbb{C}$-vector space structure of $Ω$ unchanged. One can then apply the usual algebraic tools to study differential operators: finding generators of the kernel and image, computing a Hilbert polynomial of these modules, etc. Taking differential operators arising from a distinguished family of derivations, we are able to classify which of them allow such deformations on $Ω$. Finally we give examples of differential operators and the deformations that they induce.

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arXiv2016

Unfoldings and Deformations of Rational and Logarithmic Foliations

We study codimension one foliations in projective space \PP^n over \CC by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type. For a differential form ωdefining a codimension one foliation, we present a graded module \UU(ω), related to the first order unfoldings of ω. If ωis a generic form of rational or logarithmic type, as a first application of the construction of \UU(ω), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of ω, in terms of a Hilbert polynomial associated to \UU(ω). We review the notion of regularity of ωin terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, ωis regular if and only if every unfolding is trivial up to isomorphism.

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