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The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree 5 and a ball quotient. (English) Zbl 1279.14049
Let \(S\) be a smooth projective surface with ample canonical bundle and let \(D\) be a reduced simple normal crossing divisor on \(S\) (maybe \(D=0\)). It is a well known fact that the (logarithmic) Chern numbers \(\overline{c}^2_1\) and \(\overline{c}_2\) of \(S'=S-D\) satisfy \[ \overline{c}^2_1\leq3\overline{c}_2\,, \] and the equality holds if and only if \(S'\) is a ball quotient. Few constructions of surfaces with Chern ratio \(\frac{\overline{c}^2_1}{\overline{c}_2}=3\) are known.
In the paper under review, the author considers the Fano surface \(S\) (which is a surface of general type) parameterizing lines on the Fermat cubic threefold. He shows that the divisor \(D\) consists of \(12\) disjoint elliptic curves and that \(S'\) is a ball quotient with log Chern numbers \(\overline{c}_1^2=3\overline{c}_2=3^4\).
Using a result of [M. Namba, Branched coverings and algebraic functions. Pitman Research Notes in Mathematics Series, 161. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons Ltd. (1987; Zbl 0706.14017)] the author proves that there exist a degree \(3^4\) ramified cover \(\eta: S\rightarrow\mathcal{H}_1\) branched with order 3 over the ten \((-1)\)-curves of \(\mathcal{H}_1\), the del Pezzo surface of degree 5. Moreover, there exists an étale map \(\kappa: \mathcal{H}_3\rightarrow S\) that is a quotient of \(\mathcal{H}_3\) by an automorphism of order 3, where \(\eta_3: \mathcal{H}_3\rightarrow\mathcal{H}_1\) is a degree \(3^5\) cover branched over the ten \((-1)\)-curves of \(\mathcal{H}_1\) with order \(3\) constructed by F. Hirzebruch [Progr. Math. 36, 113–140 (1983; Zbl 0527.14033)].
In analogy with a result of T. Yamazaki and M. Yoshida [Math. Ann. 266, 421–431 (1984; Zbl 0513.14008)], the author shows that the surface \(\mathcal{T}=\kappa^{-1}S' \subset \mathcal{H}_3\) is a ball quotient: \(\mathcal{T}\cong \mathbb{B}_2/\Lambda\), where \(\Lambda\) is the commutator group of the Deligne-Mostow lattice associated to the \(5\)-tuple \((1/3,1/3,1/3,1/3,2/3)\) [P. Deligne and G. D. Mostow, Publ. Math., Inst. Hautes Étud. Sci. 63, 5–89 (1986; Zbl 0615.22008)].

MSC:
14J29 Surfaces of general type
14J25 Special surfaces
22E40 Discrete subgroups of Lie groups
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