We give an explicit construction of families of Dm-equivariant polynomial vector fields in R possessing a codimension-one homoclinic cycle. The homoclinic cycle consist of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m a multiple of f...

Let $X$ be a smooth projective variety. We show that the map sends codimension one distribution on to its singular scheme is morphism from moduli space of distributions into Hilbert scheme. describe fibers and, when $X = \mathbb{P}^n$, compute them via syzygies. As an application, we spaces degree 1 $\mathbb{P}^3$. also give minimal graded free resolution for ideal generic

In these notes, we present a general framework to compute the codimension one part of the elimination ideal of a system of homogeneous polynomials. It is based on the computation of the so-called MacRae’s invariants that we will obtain by means of determinants of complexes. Our approach mostly uses tools from commutative algebra. We begin with some basics on elimination theory and then introduc...

We show that if Λ is a codimension-one hyperbolic attractor for a Cr diffeomorphism f , where 2 ≤ r ≤ ∞, and f is not Anosov, then there is a neighborhood U of f in Diffr(M) and an open and dense set V of U such that any g ∈ V has a trivial centralizer on the basin of attraction for Λ.

We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on it...

To any d-web of codimension one on a holomorphic n-dimensional manifold M (d > n), we associate an analytic subset S of M . We call ordinary the webs for which S has a dimension at most n− 1 or is empty. This condition is generically satisfied. We prove that the rank of a ordinary d-web has an upper-bound π′(n, d) which, for n ≥ 3, is strictly smaller than the bound π(n, d) of Castelnuovo. This...

Abstract Based on the celebrated result zeros of holomorphic 1-forms complex varieties general type by Popa and Schnell, we study $n$-dimensional Kodaira dimension $n-1$. We show that a minimal smooth projective variety $X$ $\kappa (X)=\dim X-1$ admits 1-form without zero if only there is morphism from to an elliptic curve. Furthermore, for (not necessarily minimal) codimension one, give struct...

We introduce the concept of topological finite-determinacy for germs analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We generalize classical results of Thom and Varchenko stating the following: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged un...