Related papers: Bergman spaces on algebraic curves
The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker…
We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.
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…
The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on…
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.…
In the first part of the paper Beilinson's theorem on the bounded derived category of coherent sheaves on P^n is extended to weighted projective spaces in a rather explicit form. To this purpose the usual category of coherent sheaves is…
In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector…
We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's…
We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces…
On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is…
We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$,…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection…
We generalize the inequality being a counterpart of the several complex variables version of the Suita conjecture. For this aim higher order generalizations of the Bergman kernel are introduced. As a corollary some new partial results on…
For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…
This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their…
Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an…
We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the…
The paper presents an analog of the old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can…