Related papers: Geometry of division rings

Let $S$ be an abelian surface over an algebraically closed field $k$ with characteristic different from $2$ and $3$, and $\mathcal{L}$ a symmetric ample line bundle defining a polarisation of type $(1,3)$. Then the linear system…

Let $S \to C$ be a Kodaira fibration. Here we show that whether or not the algebraic surface $S$ is defined over a number field depends only on the biholomorphic class of its universal cover.

For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least…

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic…

This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge…

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.

We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including…

We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a…

Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3 over an uncountable algebraically closed field. Suppose that the graded quotient ring of R is a skew-Laurent ring over a field; we say that R is a birationally…

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of $\mathbb{P}^2$ branched along a smooth sextic curve, we give a bound for the degree…

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…

We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of…

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

We prove a new bound for the number of connected components of a real regular elliptic surface with a real section and we show the sharpness of this bound. Furthermore, all possible values for the Betti numbers of such a surface are…

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…

We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general…

In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial…