Related papers: Arcs in finite projective spaces

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…

Let K=Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct…

We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…

In this expository article, we follow the work of Langer to prove the boundedness of the moduli space of semistable torsion-free sheaves over a projective variety, in any characteristic.

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over…

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

We study the space of arcs on a singularity of the form xy=f(z_1,..., z_n) and prove 2 main results. (i) The number of irreducible components equals the multiplicity of f minus 1. (ii) If n>1 and the leading homogeneous term of f is not a…

In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points and tangents, and then shows that it actually coincides with the…

We prove Sklar's theorem in infinite dimensions via a topological argument and the notion of inverse systems.

Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $\psi \colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot…

We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…

We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…

Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit…

Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define the super analog of projective modules and explore this link in the case of some particular super geometric…

We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.

We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this…

We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both…

This thesis establishes new quantitative records in several problems of incidence geometry and growth. After the necessary background in Chapters 1, 2 and 3, the following results are proven. Chapter 4 gives new results in the incidence…

We prove Libgober's divisibility relations for Oka and Alexander polynomials of symplectic curves in the complex projective plane. Along the way, we give new proofs of the divisibility relations for the Alexander polynomials of complex…