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\(K3\) surfaces with symplectic group actions. (English) Zbl 1055.14043
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 167-182 (2003).
Let \(k\) be a field of positive characteristic, and let \(X\) be a \(K3\) surface over \(k\). A finite group action \(G\) on \(X\) is called symplectic if every element of \(G\) fixes the nowhere vanishing \(2\)-form on \(X\). The paper under review looks at symplectic group actions on \(K3\) surfaces defined over a field \(k\) of positive characteristic, and discuss their influences on the quantities that exist exclusively in positive characteristic, e.g., the height of formal Brauer groups, Artin invariants of \(K3\) surfaces. The main results established here are that the height of the formal Brauer group and the Artin invariant of a \(K3\) surface are invariant under symplectic group actions. Special attention is given to the \(K3\) surfaces, which are one-parameter deformations of \(K3\) surfaces in weighted projective \(3\)-spaces. A typical example of such a surface is a deformation of a weighted diagonal \(K3\) surface, and is defined by the equation of the form \(x_0^{d_0}+x_1^{d_1}+x_2^{d_2}+x_3^{d_3}-\lambda x_0x_1x_2x_3=0\) with a parameter \(\lambda\) in a weighted projective \(3\)-space. Picard numbers, the height of formal Brauer groups, and Artin invariants are computed for these \(K3\) surfaces. Moreover, the Tate conjecture is proved for (some) of these \(K3\) surfaces over a finite field \(k\). Also a formula counting the number of rational points over a finite field of \(q\)-elements on these \(K3\) surfaces is given.
For the entire collection see [Zbl 1022.00014].

14J28 \(K3\) surfaces and Enriques surfaces
11G25 Varieties over finite and local fields
14F22 Brauer groups of schemes
14L30 Group actions on varieties or schemes (quotients)
14C22 Picard groups