Related papers: Weak Approximation for General Degree Two del Pezz…

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…

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…

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…

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 study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite…

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…

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

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type…

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves…

In this article, we consider weak del Pezzo surfaces defined over a finite field, and their associated, singular, anticanonical models. We first define arithmetic types for such surfaces, by considering the Frobenius actions on their Picard…

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…

We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational…

I construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity q>0. Such things can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo…

Let $K=k(C)$ be the function field of a curve over a field $k$ and let $X$ be a smooth, projective, separably rationally connected $K$-variety with $X(K)\neq\emptyset$. Under the assumption that $X$ admits a smooth projective model $\pi:…

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…

Hirschfeld classified split del Pezzo surfaces of degree at least three whose points are all contained on the lines in the surface. We continue his work and begin the classification of split degree two del Pezzo surfaces over finite fields…

We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root…

We find normal forms for del Pezzo surfaces of degree $2$ over algebraically closed fields of characteristic $2$. For each normal form, we describe the structure of the group of automorphisms of the surface. In particular, we classify all…

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…