Related papers: On Birational Transformations of Pairs in the Comp…

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y). We discuss various methods to construct such…

We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between…

The paper reviews recent developments in the study of Alexander invariants of quasi-projective manifolds using methods of singularity theory. Several results in topology of the complements to singular plane curves and hypersurfaces in…

We classify projective plane nonsingular curves admitting a 3-term presentation; they exist in any degree, generally constitute 5 birational families and are defined over rational numbers. The Belyi functions on all these curves are…

This article studies the possible degenerations of plane Cremona transformations of some degree into maps of smaller degree.

Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…

Jordan analytic curves which are invariant under rational functions are studied

The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}^d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq…

This survey is an invitation to recent developments in higher dimensional birational geometry.

Properties of the recently reported homogeneous Hilbert curves are deduced and reported. The nature of the affine transformations involved in the construction of the Hilbert curves is explored. The analytical representation of proper and…

This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way…

The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…

We study the congruence of bitangent lines of an irreducible surface in the 3-dimensional projective space in arbitrary characteristic, with special attention to quartic surfaces with rational double points and, in particular, Kummer…

Let $K$ be an algebraically closed field of arbitrary characteristic and let $X$ be an irreducible projective variety over $K$. Let $G\subseteq\text{Bir}(X)$ be a bounded-degree subgroup. We prove that there exists an irreducible projective…

We construct a sequence of explicit blow-ups and blow-downs on irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover this resolves a birational map given by Landsberg…

This is a survey article devoted to the study of real structures on complex algebraic varieties endowed with a reductive group action.

This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…

I show that the cohomology of the generic points of algebraic complex varieties becomes {\sl stable} birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'.