Numerically trivial automorphisms of Enriques surfaces in characteristic 2. (English) Zbl 1432.14032
Let \(S\) be an Enriques surface over an algebraically closed field \(k\) of characteristic \(p\ge 0\). An automorphism of \(S\) is called cohomologically trivial (resp. numerically trivial) if it acts identically on \(\mathrm{NS}(S)\) (resp. on \(\mathrm{Num}(S)=\mathrm{NS}(S)/\langle K_S\rangle\)). The groups of such automorphisms are denoted by \(\mathrm{Aut}_{\operatorname{ct}}(S)\) and \(\mathrm{Aut}_{\operatorname{nt}}(S)\), respectively. A complete classification of the possible groups of cohomologically or numerically trivial automorphisms on complex Enriques surfaces is due to S. Mukai [Kyoto J. Math. 50, No. 4, 889–902 (2010; Zbl 1207.14038); with Y. Namikawa, Invent. Math. 77, 383–397 (1984; Zbl 0559.14038)].
The present paper extends such results to algebraically closed fields of any characteristic \(p\ge 0\). In particular, the authors prove the following assertions.
If \(p\ne 2\), then the main result of Mukai and Namikawa still holds: \(\mathrm{Aut}_{\operatorname{ct}}(S)\) has order \(\le 2\), while \(\mathrm{Aut}_{\operatorname{nt}}(S)\cong \mathbb{Z}/2^a\mathbb{Z}\) for some \(a\le 2\).
If \(p=2\) and \(K_S\ne 0\) (\(S\) is classical), then \(\mathrm{Aut}_{\operatorname{ct}}(S)\) has order \(\le 2\) and \(\mathrm{Aut}_{\operatorname{nt}}(S)\cong (\mathbb{Z}/2\mathbb{Z})^a\) for some \(a\le 2\), unless \(S\) has a configuration of \((-2)\)-curves of type \(\widetilde{E}_8\).
If \(p=2\) and \(K_S=0\), then \(\mathrm{Aut}_{\operatorname{ct}}(S)=\mathrm{Aut}_{\operatorname{nt}}(S)\) has order \(\le 2\), unless \(S\) is supersingular with one of five types of exceptional configurations of \((-2)\)-curves.
The main tool of the proof are bielliptic maps. They are maps \(S\rightarrow \mathbb{P}^4\) of degree \(2\) onto a weak del Pezzo surface of degree \(4\), and they are induced by the linear systems \(|2F_1+2F_2|\), where \(F_1,F_2\) are half-fibers of genus one fibrations on \(S\) with \(F_1F_2=1\).
The present paper extends such results to algebraically closed fields of any characteristic \(p\ge 0\). In particular, the authors prove the following assertions.
If \(p\ne 2\), then the main result of Mukai and Namikawa still holds: \(\mathrm{Aut}_{\operatorname{ct}}(S)\) has order \(\le 2\), while \(\mathrm{Aut}_{\operatorname{nt}}(S)\cong \mathbb{Z}/2^a\mathbb{Z}\) for some \(a\le 2\).
If \(p=2\) and \(K_S\ne 0\) (\(S\) is classical), then \(\mathrm{Aut}_{\operatorname{ct}}(S)\) has order \(\le 2\) and \(\mathrm{Aut}_{\operatorname{nt}}(S)\cong (\mathbb{Z}/2\mathbb{Z})^a\) for some \(a\le 2\), unless \(S\) has a configuration of \((-2)\)-curves of type \(\widetilde{E}_8\).
If \(p=2\) and \(K_S=0\), then \(\mathrm{Aut}_{\operatorname{ct}}(S)=\mathrm{Aut}_{\operatorname{nt}}(S)\) has order \(\le 2\), unless \(S\) is supersingular with one of five types of exceptional configurations of \((-2)\)-curves.
The main tool of the proof are bielliptic maps. They are maps \(S\rightarrow \mathbb{P}^4\) of degree \(2\) onto a weak del Pezzo surface of degree \(4\), and they are induced by the linear systems \(|2F_1+2F_2|\), where \(F_1,F_2\) are half-fibers of genus one fibrations on \(S\) with \(F_1F_2=1\).
Reviewer: Giacomo Mezzedimi (Hannover)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
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