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We present a method for sampling points from an algebraic manifold, either affine or projective, defined over a local field, with a prescribed probability distribution. Inspired by the work of Breiding and Marigliano on sampling real…
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.
Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this…
We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index)…
We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves…
This article is devoted to studying complex algebraic sets under (global) blow-spherical equivalence. The main results of this article are complete classifications of complex algebraic curves. Firstly, we present a complete classification…
We obtain sufficient and necessary conditions (in terms of positive singular metrics on an associated line bundle) for a positive divisor $D$ on a projective algebraic variety $X$ to be attracting for a holomorphic map $f:X \mapsto X$.
We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
It is well-known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane,…
We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras.…
We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the…
We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use…
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.
The aim of the paper is to discuss the relations between the three kinds of objects named in the title. In a sense, this is a survey of such relations; however, some new directions are also considered. This relates, especially, to sections…