Related papers: Rational Degree Algebraic Geometry

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…

We determine explicit quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centers and block diagonal forms {of these algebras.} In the case where $q$ is {an arbitrary} root of unity, this further…

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational…

In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and…

Algebraic-geometric codes can be constructed by evaluating a certain set of functions on a set of distinct rational points of an algebraic curve. The set of functions that are evaluated is the linear space of a given divisor or,…

Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space equal to the projectivization of the noncommutative symmetric algebra of a k-central two -sided…

It is a fairly known fact that most of the algebras appearing in the theory of rings of differential operators, quantized algebras of different kinds (including many quantum groups), regular algebras in projective non-commutative geometry,…

We present bounds for the geometric degree of the tangent bundle and the tangential variety of a smooth affine algebraic variety $V$ in terms of the geometric degree of $V$. We first analyze the case of curves, showing an explicit relation…

An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…

Line bundles of rational degree are defined using Perfectoid spaces, and their co-homology computed via standard \v{C}ech complex along with Kunneth formula. A new concept of `braided dimension' is introduced, which helps convert the curse…

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…

This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively…

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras $\mathcal{B}(p;q, 0)$ (over a ring of polynomials in one variable) are…

Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle…

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…

This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three.…