Related papers: Remarks on rational vector fields on $\mathbb C\ma…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are…
Let $D$ be an effective divisor on a smooth projective variety $X$ over an algebraically closed field $k$ of characteristic $0$. We show that there is a one-to-one correspondence between the class of orthogonal (respectively, symplectic)…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere $S^2$ is proved. A classification of these families up to moderate equivalence in neighborhoods of their large…
In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\C^2$. The general problem is also reduced to the study of the combinatorics of…
Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group…
We define generalized vector fields, and contraction and Lie derivatives with respect to them. Generalized commutators are also defined.
We generalize double bracket vector fields, originally defined on semisimple Lie algebras, to Poisson manifolds equipped with a pseudo-Riemannian metric by utilizing a symmetric contravariant 2-tensor field. We extend the normal metric on…
We study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with the property that the field of definition of one of their iterates drops in degree (over a given…
In this paper we study a general class of conics starting from a quotient field. We give a group structure over these conics generalizing the construction of a group over the Pell hyperbola. Furthermore, we generalize the definition of…
We study the general rational trigonometry of a tetrahedron, based on quadrances, spreads and solid spreads, using vector products associated to an arbitrary symmetric bilinear form over a general field, not of characteristic two. This…
Generators for the module of vector fields liftable over corank 1 stable complex analytic maps from an n-manifold to an (n+1)-manifold are found. This is applied to the classification of the singularities occuring in generic one-parameter…
We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.
In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived…
We define and study a certain category of vector bundles on a p-adic curve to which we can associate in a functorial way finite dimensional p-adic representations of the geometric fundamental group. Among other things we investigate two…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
For a finite dimensional vector space equipped with a $\mathbb C$-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.