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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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References:
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