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Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite-dimensional motive, in the sense of Kimura. We also establish the Kuga–Satake Hodge conjecture for S, as well as Voisin's conjecture concerning zero-cycles. As a consequence, we obtain Kimura finite-dimensionality, the Kuga–Satake Hodge conjecture, and Voisin's conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism. [ABSTRACT FROM AUTHOR]

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]

A Shioda–Inose structure is a geometric construction which associates to an Abelian surface a projective K3 surface in such a way that their transcendental lattices are isometric. This geometric construction was described by Morrison by considering special symplectic involutions on the K3 surfaces. After Morrison several authors provided explicit examples. The aim of this paper is to generalize Morrison's results and some of the known examples to an analogous geometric construction involving not involutions, but order 3 automorphisms. Therefore, we define generalized Shioda–Inose structures of order 3, we identify the K3 surfaces and the Abelian surfaces which appear in these structures and we provide explicit examples. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface and let g be a non-symplectic involution of X such that the fixed points set contains a curve of genus 8 or more. In this paper, we show that the quotient space X/〈g〉 is determined by the fixed points set and the action of g on rational curves on X. [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]

The automorphism group of a K3 surface with Picard number two is either the infinite cyclic group or the infinite dihedral group, if it is infinite. In this paper, we determine some conditions for a K3 surface of Picard number two to have the infinite dihedral automorphism group. [ABSTRACT FROM AUTHOR]

Given d ∈ ℕ , we prove that all smooth K3 surfaces (over any field of characteristic p ≠ 2 , 3) of degree greater than 8 4 d 2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d ≥ 3 , we also construct K3 surfaces of any degree greater than 4 d (d + 1) with 24 rational curves of degree exactly d , thus attaining the above bounds. [ABSTRACT FROM AUTHOR]

In this paper we study non-symplectic automorphisms of odd order on K3 surfaces which are not purely. In particular we shall describe fixed loci of non-symplectic automorphisms of order 15 and 21. [ABSTRACT FROM AUTHOR]

We study cyclic quotient singularities given by automorphisms of maximum order on K3 surfaces. In particular, we describe fixed loci of such automorphisms, and provide types of these singularities. [ABSTRACT FROM AUTHOR]

We determine the automorphism groups of some K3 surfaces of Picard number 2. These are constructed by generalizing the Cayley–Oguiso's automorphism. These automorphisms induce natural automorphisms on the Hilbert schemes of length 2 of such K3 surfaces. Moreover, on such K3 surfaces, very ample divisors inducing isomorphisms with a quartic hypersurface define Beauville involutions of the Hilbert schemes. We also determine the relations between the natural automorphisms and the Beauville involutions on the Hilbert schemes. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface, and g be a primitive non-symplectic automorphism of X of order 3, 4, or 6. We write the quotient space as X / 〈 g 〉 and the quotient morphism as p : X → X / 〈 g 〉 . In this paper, we study curves on X obtained by pulling back curves on X / 〈 g 〉 by p. As a result, we show that if X / 〈 g 〉 is smooth, then X / 〈 g 〉 is the projective plane ℙ 2 or Hirzebruch surfaces F l of degree l = 0 , 1 , 2 , 3 , 4 , 6 , or 12. In addition, we show that if X / 〈 g 〉 is smooth, then X / 〈 g 〉 is determined by the set of fixed points of g and the action of g on rational curves on X. [ABSTRACT FROM AUTHOR]

We construct automorphisms of positive entropy of K3 surfaces of Picard number 4 with certain Salem numbers. We also prove that there is a fixed point free automorphism of positive entropy on a K3 surface of Picard number 4 with Salem degree 4. [ABSTRACT FROM AUTHOR]

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor. Furthermore, he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface. Subsequently, he studies the same theme for Enriques surfaces. [ABSTRACT FROM AUTHOR]

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p-adic method, we count the number of points on some K3 surfaces over the field F p , for all primes p < 10 8 . [ABSTRACT FROM AUTHOR]

We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations. [ABSTRACT FROM AUTHOR]

The three pencils of K3 surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry–Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements. [ABSTRACT FROM AUTHOR]

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x , y , and z , using creative telescoping, yielding modular forms expressed as pullbacked 2 F 1 hypergeometric functions, can be obtained much more efficiently by calculating the j -invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p = x y z . In other cases where creative telescoping yields pullbacked 2 F 1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2 F 1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. [ABSTRACT FROM AUTHOR]

Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollár). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale-Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale-Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic Kähler orbifolds to the algebro-geometric conjecture by Muñoz, Rojo and Tralle that claims that for an algebraic surface with b1 = 0 and b2 > 1 there cannot be b2 smooth disjoint complex curves of genus g > 0 spanning the (rational) homology. [ABSTRACT FROM AUTHOR]

Fix a K3 lattice of rank 2 and a big and nef divisor L 2 that is suitably positive. We prove that the generic polarised K3 surface has an integral nodal rational curve in the linear system |L|, in particular strengthening previous work of the first-named author. The technique is by degeneration and also works for many lattices of higher rank. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface and L be an ample line bundle on it. In this article, we will give an alternative and elementary proof of Lelli-Chiesa’s theorem in the case of r = 2 . More precisely, we will prove that under certain conditions the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system |L|. Using Lelli-Chiesa’s theorem for r ≥ 3 , we also extend Lelli-Chiesa’s theorem in the case of r = 2 in weaker condition. [ABSTRACT FROM AUTHOR]

The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of 96 % of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective K3 surface over the complex numbers and let C be an ample curve on X. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle EC,A associated to a line bundle A on C such that |A| is a pencil on C and computes the Clifford index of C. We give a necessary and sufficient condition for EC,A to be semistable. [ABSTRACT FROM AUTHOR]

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves. [ABSTRACT FROM AUTHOR]

In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere packings in two and three dimensions. In this work, we investigate the generalization in seven and eight dimensions and show that they too have many of the properties of those in lower dimensions. In particular, the hyperspheres are tangent or do not intersect; they fill the hyperspace; the object includes a maximal cluster of mutually tangent hyperspheres; and there exists a perspective where all hyperspheres in the object have integer curvatures. [ABSTRACT FROM AUTHOR]

The extensive computer-aided search applied in Bena et al. (The tadpole problem, 2020) to find the minimal charge sourced by the fluxes that stabilize all the (flux-stabilizable) moduli of a smooth K3 × K3 compactification uses differential evolutionary algorithms supplemented by local searches. We present these algorithms in detail and show that they can also solve our minimization problem for other lattices. Our results support the Tadpole Conjecture: The minimal charge grows linearly with the dimension of the lattice and, for K3 × K3, this charge is larger than allowed by tadpole cancellation. Even if we are faced with an NP-hard lattice-reduction problem at every step in the minimization process, we find that differential evolution is a good technique for identifying the regions of the landscape where the fluxes with the lowest tadpole can be found. We then design a "Spider Algorithm," which is very efficient at exploring these regions and producing large numbers of minimal-tadpole configurations. [ABSTRACT FROM AUTHOR]

Conjecturally, Fano varieties of K3 type admit a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti–Mongardi. This has interesting consequences for the Chow ring of these varieties. [ABSTRACT FROM AUTHOR]

We prove that certain Fano fourfolds of K3 type constructed by Fatighenti–Mongardi have a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.

In this paper we study non-symplectic automorphisms of order 6 on K3 surfaces which are not purely. In particular we shall describe their fixed loci. [ABSTRACT FROM AUTHOR]

A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves. [ABSTRACT FROM AUTHOR]

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K3 hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families. [ABSTRACT FROM AUTHOR]

Let (X, H) be a polarized K3 surface with Pic (X) = Z H , and let C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d (≥ 5) smooth plane curve is d - 4 , which is the same as the Clifford index of the curve. [ABSTRACT FROM AUTHOR]

General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

In this article, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces. [ABSTRACT FROM AUTHOR]

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization. [ABSTRACT FROM AUTHOR]

We prove that the projectivized cotangent bundles of smooth quadrics of dimensions three and four are (p - 1) -th Frobenius split when p > 10 . Besides, we show that the cotangent bundles of certain ordinary elliptic K3 surfaces are not Frobenius split. [ABSTRACT FROM AUTHOR]

We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space. [ABSTRACT FROM AUTHOR]

We give more detail to our examples in [1] of K3 surfaces over which have an infinite automorphism group that preserves some elliptic pencil of the K3 surface. [ABSTRACT FROM AUTHOR]

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 3, which is supersingular. The specialization induces an embedding of the Néron–Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point-free involution that gives rise to the Enriques surface of type IV in Nikulin–Kondo–Martin's classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3. [ABSTRACT FROM AUTHOR]

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if p≢1 mod 12. Our methods combine different approaches such as quotients by the group scheme αp, Kummer surfaces, and automorphisms of hyperelliptic curves. [ABSTRACT FROM AUTHOR]

We report on our results concerning the distribution of the geometric Picard ranks of K3 surfaces under reduction modulo various primes. In the situation that rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to (- 1) . [ABSTRACT FROM AUTHOR]

We characterize the standard smooth structure of K3 among all smooth structures by the existence of a certain symplectic symmetry of order 96. [ABSTRACT FROM AUTHOR]

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over Q together with a 3-torsion Brauer class α that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y. Motivated by Hodge theory, the pair (Y, α) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for α. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold X (and hence the fibers) over Q3 and local solubility at all other primes. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface and let H be a very ample line bundle on X of sectional genus g ≤ 9. In this paper, we characterize the destabilizing sheaf of the Lazarsfeld-Mukai bundle EC,Z of rank 2 associated with a smooth curve C ∈ |H| and a base point free divisor Z on C with h0(OC(Z)) = 2, in the case where it is not H-slope stable. [ABSTRACT FROM AUTHOR]