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We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a K 3 surface over a number field k. In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character [3, Definition 2.4.6] entirely in terms of the endomorphism field and the geometric Picard rank. [ABSTRACT FROM AUTHOR]

This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant −1. As an application, we show that the logarithm of every Salem number of degree 20 is realized as the topological entropy of an automorphism of a nonprojective K3 surface. [ABSTRACT FROM AUTHOR]

An automorphism of order n of a K3 surface is called purely non-symplectic if it multiplies the holomorphic symplectic form by a primitive n -th root of unity. We give the classification of purely non-symplectic automorphisms with φ (n) ≥ 12 where φ denotes the Euler totient function. [ABSTRACT FROM AUTHOR]

In this paper we study the moduli spaces of nodal sextic curves. We realize each irreducible component of the GIT space of sextic curves with given number of nodes as an open subspace of type IV arithmetic quotients. We then focus on the compactifications of the moduli spaces, one side is the geometric (GIT) compactifications, the other side is the Hodge theoretic compactifications such as Looijenga compactifications and Baily-Borel compactifications. The main result is the isomorphism between GIT and Looijenga compactifications. Some examples are closely related to del Pezzo surfaces. We also extend our results to moduli of nodal sextic curves with specified symmetry. [ABSTRACT FROM AUTHOR]

We construct examples of hyperKähler manifolds of Picard number two with automorphisms of positive entropy via derived automorphisms of K3 surfaces of Picard number one. Our hyperKähler manifolds are constructed as moduli spaces of Bridgeland stable objects in derived categories of K3 surfaces. Then automorphisms of positive entropy are induced by derived automorphisms of positive entropy on K3 surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study A 1 curves on log K3 surfaces. We classify all genuine log K3 surfaces of type II which admit countably infinite A 1 curves. [ABSTRACT FROM AUTHOR]

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi–Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi–Yau threefolds, including their Hodge numbers and deformation theory. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface, let C be a smooth curve on X , and let Z be a base point free pencil on C . Then, the Lazarsfeld–Mukai bundle E C , Z of rank 2 associated with C and Z is given by an extension of the torsion free sheaf J Z ⊗ O X ( C ) by O X , where J Z is the ideal sheaf of Z in X . We can see that if C is very ample as a divisor on X , E C , Z is an ACM bundle with respect to O X ( C ) . In this paper, by using this fact, we will characterize a necessary condition for E C , Z to be given by an extension of two line bundles on X , by ACM line bundles with respect to O X ( C ) . [ABSTRACT FROM AUTHOR]

We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if A and A ′ are Morita equivalent Azumaya algebras of degree r then 2 r divides c 2 ( A ) − c 2 ( A ′ ) . In particular this implies that if A is an Azumaya algebra on a K3 surface and c 2 ( A ) is within 2 r of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non-empty, is a proper algebraic space. [ABSTRACT FROM AUTHOR]

Abstract: We treat non-symplectic automorphisms on K3 surfaces which act trivially on the Néron–Severi lattice. In this paper, we classify non-symplectic automorphisms of prime-power order, especially 2-power order on K3 surfaces, i.e., we describe their fixed locus. [Copyright &y& Elsevier]

Abstract: We construct an Enriques surface X over with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer–Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle. [Copyright &y& Elsevier]

Abstract: We study Hessian K3 surfaces of non-Sylvester form. They are obtained as toric hypersurfaces, and their periods satisfy the Lauricella''s hypergeometric differential equation . The period domain is the Siegel upper half-space of degree 2. We construct modular forms on it using results of Ibukiyama. [Copyright &y& Elsevier]

Abstract: In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is . [ABSTRACT FROM AUTHOR]

Abstract: We show that Wehler K3 surfaces with Picard number three, which are the focus of the previous paper by Arthur Baragar, do indeed exist over the rational numbers. [ABSTRACT FROM AUTHOR]

Abstract: We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981. [Copyright &y& Elsevier]

Abstract: This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Néron–Severi group. Complementing a result by Vorontsov and Kondō, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on relations with mirror symmetry. [Copyright &y& Elsevier]

Abstract: For each positive integer , we present a tessellation of that can be lifted, through the branched covering, to a symmetric tessellation of the Fermat surface (a 4-manifold) of degree in . The process is systematic and symbolically algebraic. Each four-cell in the tessellation is bounded by four pentahedrons, and each pentahedron has four triangular faces and one quadrilateral face. Graphically, one can produce the entire surface from one single four-cell using translations generated by permutations and phase multiplications of the homogeneous coordinates of . Note that the tessellation of the Fermat surface of degree 4, a K3 surface, has exactly 24 vertices. [Copyright &y& Elsevier]

Abstract: For a K3 surface X and its bounded derived category of coherent sheaves , we have the notion of stability conditions on in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in with a fixed numerical class and a phase is represented by an Artin stack of finite type over . Then following D. Joyce''s work, we introduce the invariants counting semistable objects in , and show that the invariants are independent of a choice of a stability condition. [Copyright &y& Elsevier]

Abstract: We develop an algorithm to determine a one-parameter family of elliptic curves associated to a one-parameter family of K3 surfaces with generic Picard number 19 by a Shioda–Inose structure. The family of elliptic curves is determined up to an isomorphism and an isogeny. An application to a generalized congruence number problem is also discussed. [Copyright &y& Elsevier]