Related papers: Mori dream K3 surfaces of Picard number four: proj…

We study the geometry of a class of $n$-dimensional smooth projective varieties constructed by Schreieder for their noteworthy Hodge-theoretic properties. In particular, we realize Schreieder's surfaces as elliptic modular surfaces and…

We study period maps for families of $K3$ surfaces those are given by anti canonical divisors of toric varieties coming from reflexive polytopes $P_2, P_4, P_5$ and $P_r$. We obtain systems of period differential equations for these…

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 classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and points and we completely classify the seven possible configurations. If the…

The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a K3 surface with a non symplectic action of Z_4. In this paper we study the geometry of the one dimensional family of K3 surfaces…

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi…

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 construct explicit examples of $K3$ surfaces over ${\mathbb Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces…

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results…

Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the…

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated,…

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3…

We study a two-parameter family of K3 surfaces of (generic) Picard rank $18$ which is mirror to the $18$-dimensional family of elliptically fibered K3 surfaces with a section. Members of this family are given as compactifications of…

We present a method to calculate the action of the Mordell-Weil group of an elliptic K3 surface on the numerical N\'eron-Severi lattice of the K3 surface. As an application, we compute a finite generating set of the automorphism group of a…

We present three interesting projective models of the supersingular K3 surface X in characteristic 5 with Artin invariant 1. For each projective model, we determine smooth rational curves on X with the minimal degree and the projective…

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more…

We discuss the Picard group of moduli space $\mathcal{K}_g$ of quasi-polarized K3 surfaces of genus $g\leq 12$ and $g\neq 11$. In this range, $\mathcal{K}_g$ is unirational and a general element in $\mathcal{K}_g$ is a complete intersection…

The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism…

It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence…

Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this…