Related papers: Foliations by Curves with Curves as Singularities

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $\Sigma$, called section, that passes through $p$ and that meets all the…

Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ \mu(F, W)$ under blowups,…

Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in its local ring, the curves on the surface X obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these…

We consider holomorphic foliations of dimension $k>1$ and codimension $\geq 1$ in the projective space $\mathbb{P}^n$, with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive…

This paper is devoted to the study of codimension two holomorphic foliations and distributions. We prove the stability of complete intersection of codimension two distributions and foliations in the local case. Converserly we show the…

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework…

Let $\mathcal{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^n$ ($n\geq 2$) having $0$ as a weakly hyperbolic singularity. Let $T$ be a positive harmonic current directed by $\mathcal{F}$ which does…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

We use the theory of foliations to study the relative canonical divisor of a normalized inseparable base-change. Our main technical theorem states that it is linearly equivalent to a divisor with positive integer coefficients divisible by…

A singular foliation is a partition of a manifold into leaves of perhaps varying dimension. Stefan and Sussmann carried out fundamental work on singular foliations in the 1970s. We survey their contributions, show how diffeological objects…

Let ${\mathcal F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\mathbb C}^{2}$ that has a germ of irreducible holomorphic invariant curve $\gamma$. We provide a lower bound for the vanishing multiplicity…

The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed…

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by…

We prove that, if $n\geq 3$, a singular foliation $\mathcal{F}$ on $\mathbb P^n$ which can be written as pull-back, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ of degree $d\geq2$ with one or three invariant lines in general…

Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{R}^3$. We begin proving that if, for some $\epsilon>0,$ $\spec(Y)\cap…

We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $g: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^n,0)$ is again singular. This provides a generalization of previous results of…

We study the supersingular curves on Picard modular surfaces modulo a prime $p$ which is inert in the underlying quadratic imaginary field. We analyze the automorphic vector bundles in characteristic $p$, and as an application derive a…

We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and…

Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partial\Sigma\neq\varnothing$ and punctures $\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. In this paper we show that for every curve $\gamma$…

Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety…