Related papers: Separatrices for real analytic vector fields in th…

We prove that a germ of analytic vector field at $(\mathbb{R}^3,0)$ that possesses a non-constant analytic first integral has a real formal separatrix. We provide an example which shows that such a vector field does not necessarily have a…

The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the…

A separating ($M-2$)-curve is a smooth geometrically irreducible real projective curve $X$ such that $X(\mathbb{R})$ has $g-1$ connected components and $X(\mathbb{C})\setminus X(\mathbb{R})$ is disconnected. Let $T_g$ be a Teichm\"uller…

A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the…

Since the works of Krasnov and Scheiderer, there has been an interest in studying effective totally real divisors on a curve X defined over a real closed field, i.e., effective divisors supported on the real locus. Scheiderer proved that,…

Let $\mathcal{X}$ be a projective algebraic curve and denote by $\mathcal{X}^{'}$ its strict dual curve. The map $\gamma:\mathcal{X} \longrightarrow \mathcal{X}^{'}$ is called (strict) Gauss map of $\mathcal{X}$. In this manuscript, we…

We give unique analytic "normal forms" for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity of saddle-node type having a convergent formal separatrix. We specifically address the…

Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short…

We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a…

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x…

A real morphism $f$ from a real algebraic curve $X$ to $\mathbb{P}^1$ is called separating if $f^{-1}(\mathbb{R} \mathbb{P}^1) = \mathbb{R} X$. A separating morphism defines a covering $\mathbb{R} X \to \mathbb{R} \mathbb{P}^1$. Let $X_1,…

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in $({\mathbb C}^3,0)$ without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of…

Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the…

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is…

We show in this text how the most important homology equivalences of fundamental Algebraic Topology can be obtained as reductions associated to discrete vector fields. Mainly the homology equivalences whose existence -- most often…

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let $L$ be a foliation on $X$. We prove that if $L$ is closed under $p$-th powers for almost all primes $p$, then any $L$-invariant smooth formal curve is…