The Picard rank of an Enriques surface (Q1651772)

Let \(X\) be an Enriques surface (a smooth proper surface with numerically trivial canonical divisor and with second Betti number \(10\)) over an algebraically closed field of arbitrary characteristic. An important fact is that then the Picard rank of \(X\) is equal to its second Betti number \(10\). In characteristic \(0\), this follows from the Lefschetz \((1,1)\)-theorem. In positive characteristic, the known proofs of this fact are more complicated. One is by \textit{E. Bombieri} and \textit{D. Mumford} [Invent. Math. 35, 197--232 (1976; Zbl 0336.14010)] using genus one fibrations, and another is by \textit{W. E. Lang}, Math. Ann. 265, No. 1, 45--65 (1983; Zbl 0575.14032)] using lifting to characteristic \(0\). In this paper the author gives another proof of the fact. This proof relies on the Tate conjecture for \(K3\) surfaces, which is difficult and proved very recently. One first reduces the fact to the Tate conjecture for \(X\). This is then reduced to the Tate conjecture for the canonical double cover \(\tilde{X}\) of \(X\). It is known that \(\tilde{X}\) is birational to a \(K3\) surface or to \(\mathbb{P}^2\) (the latter is possible only in characteristic \(2\)). The Tate conjecture for \(\mathbb{P}^2\) is trivial, and for \(K3\) surfaces it is proved recently by \textit{K. Madapusi Pera} [Invent. Math. 201, No. 2, 625--668 (2015; Zbl 1329.14079)] and \textit{W. Kim} and \textit{K. M. Pera} [Forum Math. Sigma 4, Article ID e28, 34 p. (2016; Zbl 1362.11059)] among others.