Related papers: The density of rational points on non-singular hyp…

Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained…

Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $\P^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no…

We show that if $X$ is a projective hyperk\"ahler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a…

Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform…

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic…

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical…

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number…

We propose and compare various techiques available to produce smooth cubic hypersurfaces over a non-algebraically-closed field which have rational points but which are not stably rational over their ground field.

The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density…

We construct a non-full exceptional collection of maximal length consisting of line bundles on the blow-up of the projective plane in 10 points in general position. This provides a counterexample to a conjecture of Kuznetsov and to a…

We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish…

We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational…

We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over $\R$ but not over $\Q$. This answers an open question by C. Scheiderer posed as the second…

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We…

We prove that for any compact set B in R^d and for any epsilon >0 there is a finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum absolute value of any linear function ell: R^d --> R on X approximates the maximum…

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of…

Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude…

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the…

We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb{S}^{n+1}_{1}$ which is known as the steady state space $\mathcal{H}^{n+1}$. In this setting, under suitable constraints on the behavior of…

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…