Related papers: The Diagonal Dimension of Curves

Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible components are smooth rational curves. Taking Stein factorization of its normalization, we get a conic. We show the conic is non-split in certain cases. As an…

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More…

A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular $n$-gon with edges length $R$. C_2(R) is equivalent to having constant diameter. We show that smooth curves…

The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier…

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…

We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively…

We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

We show that there is a smooth complex projective variety, of any dimension greater than or equal to two, whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are…

We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1,…

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at…

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…

We prove that for any smooth projective variety $X$ of dimension $\geq 3$, there exists an integer $g_0=g_0(X)$, such that for any integer $g \geq g_0$, there exists a smooth curve $C$ in $X$ with $g(C)=g$.

We show various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Igusa on exponential sums.

Recently, I. Kossovskiy and R. Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in ${\mathbb C}^2$ via the dimension of the algebra of infinitesimal automorphisms. In this note, we…

A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic…

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…