Normal quintic Enriques surfaces. (English) Zbl 0955.14026

E. Stagnaro [in: Algebraic geometry – open problems, Proc. Conf., Ravello 1982, Lect. Notes Math. 997, 400-403 (1983; Zbl 0511.14020)] constructed two types of normal quintic surfaces which are birational to smooth Enriques surfaces. For the first type, the author especially proves the equivalence of (1) and (2) below for an Enriques surface \(S\).
(1) \(S\) is birational to a normal quintic \(F_5\) in \({\mathbb{P}}^3\) which has two tacnodes and two triple points in general position, where each of two tacnodal planes to \(F_5\) at two tacnodes passes through two triple points.
(2) \(S\) has a divisor \(D = e_1 + e_2 + e_3 + e_4\), where \(e_1, e_2, e_3\) are half-pencils of S with \(e_i . e_j = 1\) for \(i \neq j\) [they exist by F. R. Cossec, Math. Ann. 271, 577-600 (1985; Zbl 0541.14031)] and \(e_4 = e_2 + K_S\). Furthermore, if \(p_1, \dots, p_5\) are the intersections of \(e_1, \dots, e_4\) then \(p_1, \dots, p_5\) are distinct.
For related results see Y. Umezo [Publ. Res. Inst. Math. Sci. 33, No. 3, 359-384 (1997; Zbl 0909.14023)] where she obtains a similar result.


14J28 \(K3\) surfaces and Enriques surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
14J10 Families, moduli, classification: algebraic theory