Related papers: The Gauss map of a projective foliation

Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb…

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex…

Convergence of the Gauss resolution process for a complex singular foliation of dimension r is shown to be equivalent to finite type of a graded sheaf which is built using base (r+2) expansions of integers. As applications it is calculated…

We present a new list of irreducible components of the space of codimension two holomorphic foliations on $\mathbb P^{4}$. They are associated to the pull-back by branched rational maps of 1-dimensional foliations on $\mathbb P^3$ leaving…

We construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such…

In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products…

We consider the set $K(n,c,\rtt)$ of codimension one holomorphic foliations on $\P^n,\,\, n\geq3$, with Chern class $c$, and with a compact, connected Kupka set of radial transversal type. We will prove that foliations in this set, have a…

The $\text{PSL}(4,\mathbb{R})$ Hitchin component of a closed surface group $\pi_1(S)$ consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of $S$. We prove that the leaves of the…

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…

We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late…

Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the…

In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to…

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes…

Let $(S,\mathcal{F})$ be a foliated surface over the complex number of general type, i.e., the Kodaira dimension $\mathrm{Kod}(\mathcal{F})=2$. We study the geometry of the canonical map $\varphi$ of the foliated surface $(S,\mathcal{F})$,…

The space of holomorphic foliations of codimension one and degree $d\geq 2$ in $\mathbb{P}^n$ ($n\geq 3$) has an irreducible component whose general element can be written as a pullback $F^*\mathcal{F}$, where $\mathcal{F}$ is a general…

In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of…

In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.

We describe the topological types of leaves of generic logarithmic foliations on the complex projective plane. We prove that all leaves, except for a finite many are biholomorphic to $\mathbb{C}$ or homeomorphic to the surface known as Loch…

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian…

We complete the study of birational geometry of Fano fiber spaces $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce…