Related papers: Weak Approximation over Function Fields of Curves …

We prove weak approximation for smooth cubic hypersurfaces of dimension at least 2 defined over the function field of a complex curve.

This paper addresses weak approximation for rationally connected varieties defined over the function field of a curve, especially at places of bad reduction. Our approach entails analyzing the rational connectivity of the smooth locus of…

We prove that rationally connected varieties over the function field of a complex curve satisfy weak approximation for places of good reduction.

Let $K = K(C)$ be the function field of a smooth curve $C$. Applying the result of [Xu08], we prove that if $S/K$ is a degree one or two del Pezzo surface which can be completed to a generic family in the parametrizing space over $C$, then…

Let $k$ be a $d$-local field of characteristic 0, and let $K$ be the function field of a nice curve over $k$. We give a defect to weak approximation for reductive groups over $K$ using arithmetic dualities.

We address the problem of weak approximation for general cubic hypersurfaces defined over number fields, with arbitrary singular locus. In particular, weak approximation is established for the smooth locus of projective, geometrically…

By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there…

This is a survey of weak approximation over complex function fields, touching on the Koll'ar-Miyaoka-Mori theorem, places of good and bad reduction, the special case of rational surfaces, rationally simply connected varieties, and…

We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak weak approximation if $k$ is a number field and the Hilbert property if $k$ is Hilbertian of characteristic zero, provided that they contain a $k$-rational point…

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if…

Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point…

We give local conditions at the infinite places of a number field K ensuring that the intersection of n quadrics in projective N-space over K, N >> n, satisfies weak approximation.

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…

Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails…

Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety. We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional linear subspace $W$ of $V$ contained in $X$…

For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture…

In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll,…

We prove, via an "arithmetic surjectivity" approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for…