an:05959481
Zbl 1279.14049
Roulleau, Xavier
The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree 5 and a ball quotient
EN
Proc. Am. Math. Soc. 139, No. 10, 3405-3412 (2011).
00287718
2011
j
14J29 14J25 22E40
algebraic surfaces; ball lattices; orbifolds; Fano surfaces of cubic threefolds; degree 5 del Pezzo surface
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 [\textit{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 ??tale 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 \textit{F. Hirzebruch} [Progr. Math. 36, 113--140 (1983; Zbl 0527.14033)].
In analogy with a result of \textit{T. Yamazaki} and \textit{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 \((1/3,1/3,1/3,1/3,2/3)\) [\textit{P. Deligne} and \textit{G. D. Mostow}, Publ. Math., Inst. Hautes ??tud. Sci. 63, 5--89 (1986; Zbl 0615.22008)].
Davide Frapporti (Bayreuth)
Zbl 0706.14017; Zbl 0527.14033; Zbl 0513.14008; Zbl 0615.22008