Related papers: Extended Klein model and a bound on curves with ne…

We deal with locally free $\mathcal{O}_X$-modules with connection over a Berkovich curve $X$. As a main result we prove local and global decomposition theorems of such objects by the radii of convergence of their solutions. We also derive a…

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for…

Given a smooth affine curve X over a field k of positive characteristic, and an overconvergent F-isocrystal on X, we prove after replacing k by a finite purely inseparable extension, there exists a finite separable cover of X, the pullback…

Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to…

We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of…

In this paper, we construct the Funk-Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using…

Let $f$ be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length $\rho$ is then $o(\rho^1/2)$.…

We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK…

Let $\mathcal{S} \subset \mathbb{P}^n$ be an absolutely irreducible projective hypersurface defined over a finite field $\mathbb{F}_q$, equipped with the $\mathbb{F}_q$-Frobenius map $\Phi_q$. In this paper, we investigate irreducible…

Let $\gamma$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $\rho$ for which $\gamma$ has length at most $M…

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on…

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique)…

We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension…

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the…

Let S be a Dedekind scheme with field of functions K. We show that if X_K is a smooth connected proper curve of positive genus over K, then it admits a N\'eron model over S, i.e., a smooth separated model of finite type satisfying the usual…

For a nondegenerate irreducible curve $C$ of degree $d$ in ${\Bbb P}^r$ over ${\Bbb F}_q$ with $r \geq 3$, we prove that the number $N_q(C)$ of ${\Bbb F}_q$-points of $C$ satisfies the inequality $N_q(C) \leq (d-1)q +1$, which is known as…

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected…

We give an explicit formula for the self-intersection number of negative curves on Fermat surfaces. The formula offers us hints to either prove or disprove the Bounded Negativity Conjecture for the Fermat surfaces.

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its…

Let M be a compact, connected, orientable, irreducible 3-manifold and T an incompressible torus boundary component of M such that the pair (M,T) is not cabled. In the paper "Toroidal and Klein bottle boundary slopes" [arXiv:math/0601034] by…