Related papers: On the extendability of projective varieties: a su…
We prove a hyperplane inequality for the surface area of projection bodies.
We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…
Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover…
Every algebraic variety can be regarded as a symplectic manifold being equipped with a Kahler form. Therefore it is natural to study lagrangian geometry of any algebraic variety. We present two basic constructions which can be applied to a…
We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and…
The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.
In each of the three projective planes coordinatised by the Knuth's binary semifield $\mathbb{K}_n$ of order $2^n$ and two of its Knuth derivatives, we exhibit a new family of infinitely many translation hyperovals. In particular, when…
In this note we extend the concept height on projective spaces to that of weighted height on weighted projective spaces and show how such a height can be computed. We prove some of the basic properties of the weighted height and show how it…
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 address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.
This paper surveys hyperinterpolation, a quadrature-based approximation scheme. We cover classical results, provide examples on several domains, review recent progress on relaxed quadrature exactness, introduce methodological variants, and…
We discuss various phenomena of tangency in projective and convex geometry.
We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
The Gauss map of a projective variety $X \subset \mathbb{P}^N$ is a rational map from $X$ to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties $F$ and $Y$, we construct a…
This is a survey article on the recent developments of semipositivity, injectivity, and vanishing theorems for higher-dimensional complex projective varieties.
We provide a classification of complex projective surfaces with a holomorphic foliation whose group of birational symetries is infinite.