Related papers: Some surfaces with maximal Picard number

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 $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(\tau,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/\tau B(k)$ varies when we take a finite cover…

The rank $\rho$ of the N\'eron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\leq\rho\leq g^2=h^{1,1}.$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$…

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of…

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups.

Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(\tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of…

We work out properties of smooth projective varieties over a (not necessarily algebraically closed) field that admit collections of objects in the bounded derived category of coherent sheaves that are either full exceptional, or numerically…

We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such…

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we…

We study the relation between a certain graded part of the Jacobian ring of a projective hypersurface and a certain graded quotient for the Hodge filtration of its primitive cohomology, in the case that the hypersurface has at most isolated…

Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some…

It is proved that the degree of a morphism from a smooth projective n-fold with Picard number one to a smooth n-quadric is bounded (provided, of course, that n is at least three). Actually it has been proved some years ago, but I have never…

We study the local rigidity of projective smooth horospherical varieties of rank one and Picard number two. These varieties have been already considered by the second author in a work where their automorphism groups are computed. The…

In the present work we describe 3-dimensional complex SL_2-varieties where the generic SL_2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group…

For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber…

In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of…

Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic…

We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with…

We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory.…