Related papers: Regular Foliations and Trace Divisors

We give a concrete example of a co-existential map between continua that is not confluent.

Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred…

The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and…

Using the theory of moduli of curves, we establish various slope inequalities for general fibered surfaces. More precisely, we introduce the notion of functorial divisors on Artin stacks and prove a theorem concerning their effectiveness.…

The classes of two theta divisors on an abelian variety in the naive Grothendieck ring of varieties need not be congruent modulo the class of the affine line.

Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.

We study special linear systems called "very special" whose dimension does not satisfy a Clifford type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of…

We describe the construction of the slice fibration of a given one.

A normal partition of the edges of a cubic graph is a partition into trails (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition. We investigate this notion and give some results and problems.

We prove that localization functors of crossed modules of groups do not always admit fiberwise (or relative) versions. To do so we characterize the existence of a fiberwise localization by a certain normality condition and compute explicit…

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove…

Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We…

We invent the notion of a derived and fluid variety. Fluid variety has no proper derived variety as its subvariety. We examine some properties of fluid and derived varieties. Examples of such varieties of bands are presented.

Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute…

The method of Tur\'an in establishing the normal order for the number of prime divisors of a number is used to show that a certain class of arithmetic functions do not have a normal order.

We gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some of it is apparently not published at all.

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint…

In this paper it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C^3,0) is not a topological invariant. More precisely, we provide an example of two topologically equivalent foliations…

Motivated by the Jouanolou foliation problem, we investigate the non-algebraicity of foliations by curves on $\mathbb{P}^2_{\mathbb{C}}$. We present a criterion to show that such a foliation has no algebraic invariant curves, using a method…

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of…