Related papers: Universal torsors and Cox rings
We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.
We introduce a cohomological method to compute Cox rings of hypersurfaces in the ambient space P^1 x P^n, which is more direct than existing methods. We prove that smooth hypersurfaces defined by regular sequences of coefficients are Mori…
We investigate the distribution of rational points on singular cubic surfaces, whose coordinates have few prime factors. The key tools used are universal torsors, the circle method and results on linear equations in primes.
We investigate topological properties of torsors in algebraic geometry over adelic rings.
Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of…
We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K…
We prove that the Euler-Chow series for ruled surfaces and scrolls is rational by means of an explicit computation.
The toric surfaces for octonions and related objects are discussed.
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…
We construct explicit equations of Cartwright-Steger and related surfaces.
We study generators and relations of Cox rings of K3 surfaces of Picard number two. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and elliptic K3 surfaces.
We show that every split del Pezzo surface of degree d=5,4,3 or 2 has a universal torsor which is a dense open subset of the intersection of 6-d dilatations of the affine cone over the corresponding generalized Grassmannian G/P. Here a…
Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.
The Cox rings of del Pezzo surfaces are closely related to the Lie groups E_n. In this paper, we generalize the definition of Cox rings to G- surfaces defined by us earlier, where the Lie groups G=A_n, D_n or E_n. We show that the Cox ring…
We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of…
We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, e.g. testing finite generation or computing an explicit presentation in terms of generators and…
In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves over $\mathbb C$, and prove some results towards a matrix analogue of this problem.