Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Neron-Severi torus. Let
$mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $mathcal{T}$ over $S_L$. We prove that $mathcal{G}$ does not descend to S unless $mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $mathcal{G}$ always descend to singular del Pezzo surfaces over $mathbb{C}$ from their desingularizations.
We prove Manins conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singular
ities. Moreover, we show that one of these families does not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in the asymptotic formula. Our proofs are based on the universal torsor method, using Brions description of Cox rings of spherical varieties.
We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.
We introduce descent methods to the study of strong approximation on algebraic varieties. We apply them to two classes of varieties defined by P(t)=N_{K/k}(z): firstly for quartic extensions of number fields K/k and quadratic polynomials P(t) in one
variable, and secondly for k=Q, an arbitrary number field K and P(t) a product of linear polynomials over Q in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation.
We prove Manins conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
We prove Manins conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyres conjecture for the leading term in the asym
ptotic formula for the number of rational points of bounded height on S over number fields.
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k
]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper models of $X$. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson form
ula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
