Related papers: Transcendental obstructions to weak approximation …

Using Gauss-Manin derivatives of normal functions, we arrive at some remarkable results on the non-triviality of the transcendental regulator for $K_m$ of a very general projective algebraic manifold. Our strongest results are for the…

We study the realization spaces of $10_3$ line configurations. Answering a question posed by Sturmfels in 1991, we use elliptic surface techniques to show that realizations over $\mathbb{Q}$ are dense in those over $\mathbb{R}$ for all…

Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarized K3 surface with complex multiplication, all…

We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Spaces. We list the possible N\'eron--Severi groups of K3 surfaces with this…

Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to…

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 introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin…

We study the distribution of the Brauer group and the frequency of the Brauer--Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SL_n with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places…

We describe the equations and Gr\"obner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The…

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho\geq21-2h$ and $h\geq3$, where $\rho$ is the Picard number and $h$ is the height of the formal Brauer group. We…

We construct a quasi-canonical lifting of a $K3$ surface of finite height over a finite field of characteristic $p\geq3$. Such results are previously obtained by Nygaard-Ogus when $p\geq5$. For this purpose, we use the display-theoretic…

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic…

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute…

We consider deformations of a toroidal orbifold $T^4/Z_2$ and an orbifold of quartic in $CP^3$. In the $T^4/Z_2$ case, we construct a family of noncommutative K3 surfaces obtained via both complex and noncommutative deformations. We do this…

Let S be a projective K3 surface. It is proved that the 0-dimensional cusps of the Kahler moduli of S are in one-to-one correspondence with the twisted Fourier-Mukai partners of S. This leads to a counting formula for the 0-dimensional…

We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we…

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…

For smooth open toric varieties, we establish strong approximation off infinity with Brauer-Manin obstruction.

We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split…