\(\mathbb{Q} \ell\)-cohomology projective planes from Enriques surfaces in odd characteristic (Q6585097)

This paper gives a complete classification of \({\mathbb{Q}}_{\ell}\)-cohomology projective planes with isolated ADE-singularities and numerically trivial cannonical bundle in odd characteristic.\N\NLet \(S\) be a normal projective surface over an algebraically clsoed field \(K\) of characteristic \(p>2\) with only ADE-singularities. For a prime \(\ell\neq p\), let \(b_i(S)\) denote the Betti-numbers of \(S\) for the \(\ell\)-adic etale cohomology \(H^i_{et}(S,{\mathbb{Q}}_{\ell}).\) \(S\) is called a \({\mathbb{Q}}_{\ell}\)-cohompology projective plane if \(b_1(S)=b_3(S)=0, b_2(S)=1\). Let \(Y\) denote the minimal resolution of \(S\). Assume that \(K_Y\) is numerically trivial. Then \(Y\) is either a \(K3\) surface or an Enriques surface. In either case, \(Y\) supports a root type \(R\) of maximal rank \(b_2(Y)-1=21\), resp. \(9\) supported on smooth rational curves. \(R\) is then a direct sum of root lattices \(A, D, E\). For \(K3\) surfaces, Shimada's theorem [\textit{I. Shimada}, Math. Comput. 73, No. 248, 1989--2017 (2004; Zbl 1052.14041)] restricts the characteristic \(p\) to be in the range \(0<p\leq 19\). In contrast, Enriques surfaces present a totally different picture. This paper focuses on Enriques surfaces in odd characteristic (as the characteristic \(2\) case was dealt with in [\textit{M. Schütt}, Épijournal de Géom. Algébr., EPIGA 3, Article No. 10, 24 p. (2019; Zbl 1505.14085)]).\N\NThe main result is formulated in the following theorem.\N\NTheorem 1: There are 31 root types of rank \(9\) realized through smooth rational curves on Enriques surfaces in odd characteristic \(>5\) (28 in characteristic 434, 30 in characteristic \(5\)). For each the following hold:\N\N(i) the root types are supported on \(1\)-dimneionsional families of Enriques surfaces:\N\N(ii) the moduli spaces can have up to \(3\) different components:\N\N(iii) each family has rational base and is defined over the prime field:\N\N(iv) each family can be parametrized explicitly (e.g., Table 3).\N\NThis theorem parallels the classification result of Enriques surfaces over \({\mathbb{C}}\).\N\NThe main ingredients of proof are elliptic fibrations, Enriques involutions, extremal rational elliptic surfaces.