Related papers: Weyl group of type $E_6$ and K3 surfaces

We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space.…

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each…

We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we…

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo…

We compute the transcendental lattices of the singular K3 surfaces belonging to three pencils of K3 surfaces, namely the Ap\'ery-Fermi pencil with transcendental lattice $U\oplus \langle 12 \rangle$, the Verrill's pencil with transcendental…

We construct K\"ahler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for each $r\geq 3$, we obtain large classes of K\"ahler groups that have classifying…

We classify Jacobian elliptic fibrations on K3 surfaces with a non-symplectic automorphism $\sigma$ of order 3 according to the action of $\sigma$ on their fibres, building on work by Garbagnati and Salgado for non-symplectic involutions.…

In this paper we classify complex K3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is the disjoint…

We show that supersingular K3 surfaces in characteristic $p\geq5$ are related sequences of very special correspondences. This is not enough to conclude that they are unirational. As a byproduct, we exhibit a fibration structure on the…

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…

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic…

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…

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 classify all Jacobian elliptic fibrations on K3 surfaces with finite automorphism group. We also classify all Jacobian elliptic fibrations with finite Mordell-Weil group on K3 surfaces with infinite automorphism group and 2-elementary…

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group,…

An element in the Brauer group of a general complex projective $K3$ surface $S$ defines a sublattice of the transcendental lattice of $S$. We consider those elements of prime order for which this sublattice is Hodge-isometric to the…

Let $X$ denote a K3 surface over an arbitrary field $k$. Let $k^\text{s}$ denote a separable closure of $k$ and let $X^\text{s}$ denote the base change of $X$ to $k^\text{s}$. The action of the absolute Galois group Gal($k^\text{s}/k$) of…

Elliptic fibrations of $K3$ surfaces belonging to the Ap\'ery-Fermi pencil ($Y_k$) may have $2$ or $3$-torsion sections defining on $(Y_k)$ automorphisms $\tau$ of order $2$ or $3$. First we consider $Y_{k}/\tau$ \ for some fibrations of…

The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic…

We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal…