Related papers: Rational points of rationally simply connected var…

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…

We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that…

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G…

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

We prove that smooth non-klt toric orbifolds are separably Campana rationally connected, extending the result in the klt case. We also show that there always exists a positive characteristic in which a singular weighted projective space,…

We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from…

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we…

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre…

This is a continuation of "Rational curves on hypersurfaces of low degree", math.AG/0203088. We prove that if d^2+d+1 < n and d > 2, then for a general hypersurface X_d in P^n of degree d, for each degree e the space of rational curves of…

An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact…

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…

We consider all genus 2 curves over Q given by an equation y^2 = f(x) with f a squarefree polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200000 isomorphism classes of curves,…

Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a…

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…

In this paper we look for necessary and sufficient conditions for a genus one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus one fibration…

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I.…

Given a smooth curve $C/\mathbb{Q}$ with genus $\geq 2$, we know by Faltings' Theorem that $C(\mathbb{Q})$ is finite. Here we ask the reverse question: given a finite set of rational points $S\subseteq \mathbb{P}^n(\mathbb{Q})$, does there…

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative…