We study a pencil of K3 surfaces that appeared in the 2-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apéry-Fermi pencil, that was related to Apéry's proof of the irrationality of ζ(3) through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts. arXiv:1809.04970v2 [math.AG]
Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.Recently Keiji Oguiso showed that there exist projective K3 surfaces S with a fixed point free automorphism g of positive entropy, i.e. g * has at least one eigenvalue λ of absolute value |λ| > 1 on H 2 (S, C) (see [O]). He also described the Picard lattice of the general such surface explicitly and observed that these surfaces can be embedded into P 3 as quartic surfaces. There remained the problem of describing these quartic surfaces and their automorphism g explicitly.The aim of this paper is to provide a general method for constructing such quartic surfaces in P 3 and to describe an algorithm for finding the automorphism. Moreover, we will give an explicit example of such a surface S and automorphism g. To identify the quartic surfaces in Oguiso's construction, we observe that the Picard lattice required by Oguiso is exactly the Picard lattice of a general determinantal quartic surface, that is, the quartic equation of the surface is the determinant of a 4 × 4 matrix of linear forms.While writing the paper, we realised that such automorphisms were already described by Prof. Cayley, President of the London Mathematical Society, in his memoir on quartic surfaces, presented on February 10, 1870 ([C], §69, p.47). In fact, Cayley observed that a determinantal K3 surface S 0 ⊂ P 3 has three embeddings S i ⊂ P 3 for i = 0, 1, 2, each of which is again determinantal. The corresponding three matrices M i , which are closely related to each other, provide natural (non-linear!) maps between these three quartic surfaces. A composition of these maps is an automorphism of S 0 and we show that this automorphism is the one discovered by Oguiso.In the first section we recall Oguiso's description [O] of K3 surfaces with a fixed point free automorphism g of positive entropy. In section 1.10 we give a method that in principle allows one to give an explicit description of the automorphism. In practice, even if the K3 surface S is given as a determinantal surface in P 3 , this method is hard to use, since one needs to know certain curves of high degree on S that are not complete intersections.
We prove that every del Pezzo surface of degree 2 over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly‐Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to 2, we state sufficient conditions for a del Pezzo surface of degree 2 to be unirational.
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