Related papers: K3 surfaces with real or complex multiplication

For each integer $D\ge3$, we give a sharp bound on the number of lines contained in a smooth complex $2D$-polarized $K3$-surface in $\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in $\mathbb{P}^4$ and octics in…

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over…

Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for…

Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the N\'{e}ron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the…

In \emph{Endomorphism Algebras of Jacobians}, Ellenberg gives group theory tools to construct jacobians of curves with real multiplication. He shows the existence of curves and family of curves with real multiplication by subfields of…

In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost…

Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth $K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$ possesses an…

We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective…

We prove that for every positive integer $m\geq 18(2^{9}\cdot 3^{7})!$ and every smooth projective 3-fold of general type X defined over complex numbers, $\mid mK_{X}\mid$ gives a birational rational map from X into a projective space.

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors…

We show that on a compact complex surface all Massey products of cohomology classes in degree one vanish beyond length three. Dually, the real Malcev completion of the fundamental group is homogeneously presented by quadratic and cubic…

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field.…

It is shown that that the rank of the second fundamental form (resp. the Levi form) of a $\mathcal C^2$-smooth convex hypersurface $M$ in $\Bbb R^{n+1}$ (resp. $\Bbb C^{n+1}$) does not exceed an integer constant $k<n$ near a point $p\in M,$…

It is proven that for any topological or analytical types of isolated singular points of plane curves, there exists a non-real irreducible plane algebraic curve of degree $d$ which goes through $d^2$ real distinct points and has imaginary…

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is the symmetric group on four elements. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to…

In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $\rho (X):=\textrm{rank } \textrm{Pic } X = h_{3,2d}$…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

We study almost complex surfaces in the nearly K\"ahler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic…