In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two. Introduction K3 surfaces which are double covers of the plane ramified over a plane sextic are classical objects. In this paper we determine the automorphisms...

It is shown that the K3 spectra which refine the local rings of the moduli stack of ordinary p-primitively polarized K3 surfaces in characteristic p allow for an E∞ structure which is unique up to equivalence. This uses the E∞ obstruction theory of Goerss and Hopkins and the description of the deformation theory of such K3 surfaces in terms of their Hodge F-crystals due to Deligne and Illusie. ...

We consider N-point deformation of algebraic K3 surfaces. First, we construct two-point deformation of algebraic K3 surfaces by considering algebraic deformation of a pair of commutative algebraic K3 surfaces. In this case, the moduli space of the noncommutative deformations is of dimension 19, the same as the moduli dimension of the complex deformations of commutative algebraic K3 surfaces. Th...

The Ramsey number r(K3, Qn) is the smallest integer N such that every redblue colouring of the edges of the complete graph KN contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K3, Qn) = 2 n+1 − 1 for every n ∈ N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who prove...

Kaposi's sarcoma-associated herpesvirus (KSHV) expresses ∼20 viral microRNAs (miRNAs) in latently infected cells. We have previously shown that two of these miRNAs function as mimics of the cellular miRNAs miR-155 and miR-142-3p. Two additional KSHV miRNAs, miR-K3+1 and miR-K3, share perfect and offset 5' homology with cellular miR-23, respectively. Here, we report a single nucleotide polymorph...

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following facts: (1) For any given positive integer N , there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard groups are not isomorphic but their transcendental lattices are Hodge isometric, or equivale...

Frontotemporal dementia (FTD) is characterized by cognitive and behavioral changes and, in a significant subset of patients, Parkinsonism. Histopathologically, FTD frequently presents with tau-containing lesions, which in familial cases result from mutations in the MAPT gene encoding tau. Here we present a novel transgenic mouse strain (K3) that expresses human tau carrying the FTD mutation K36...

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 to the GromovWitten theory of 1-parameter families ofK3 surfaces. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds for O(2, 19) lattices and proven mirror transforms to determine ...

Two new bounds for multicolor Ramsey numbers are proved: R(K3,K3, C4, C4) ≥ 27 and R4(C4) ≤ 19.

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface X of finite height over a finite field k of characteristic p ≥ 5 has a quasi-canonical lifting Z to characteristic 0, and that for any such Z, the endormorphism algebra of the transcendental cycles V (Z), as a Hodge module, is a CM field over Q. The Tate conjecture for the product of...