Remarks on the bicanonical map for surfaces of general type
If a surface \(S\) of general type contains a pencil of curves of genus 2, then the bicanonical map is necessarily not birational: This is the general case for the non-birationality of the bicanonical map. In Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0663.14010), \textit{I. Reider} proved that if \(S\) is minimal, of general type, with non-birational canonical map and \(c_1(S)^2 \geq 10\), then only the general case occurs. This paper deals with the study of the special cases, i.e. of surfaces \(S\) which do not contain a pencil of curves of genus 2 and whose bicanonical map is not birational. For technical reasons only surfaces with \(p_g\geq 4\) are considered. Under this assumption a detailed description of all the surfaces for which the special case occurs. One type consists of surfaces having a pencil of hyperelliptic curves of genus 3 with a base point, which is a Weierstrass point of the hyperelliptic involution on each curve of the pencil. These surfaces are double covers of a quadratic cone in projective three-space, with a branch locus which is completely described and invariants \(p_g =6\), \(c_1 (S)^2 =9\); \(p_g =5\), \(c_1(S)^2 =7,8\); \(p_g =4\), \(c_1(S)^2 =6,7\). All these surfaces are, in fact, degenerations of the surfaces with invariants \(p_g =6\), \(c_1(S)^2 =9\). Another type consists of the double planes with a branch curve of degree 10 and these have invariants \(p_g =6\), \(c_1(S)^2 =8\). The last type consists of surfaces with invariants \(p_g =4\), \(c_1(S)^2 =8\), containing one or two pencils of hyperelliptic curves of genus 4 with two base points. These surfaces can be represented as double covers of a quadric in \(\mathbb{P}^3\) with a branch curve which is completely described. The problem of classifying the surfaces for which the bicanonical map is not birational has been considered by Du Val for regular surfaces with base point free canonical system. More recently \textit{Xiao Gang} studied the general problem of non-birationality of the bicanonical map, by taking the point of view of the projective study of the bicanonical model, thus finding a list of possibilities [see Am. J. Math. 112, No. 5, 713-737 (1990; Zbl 0722.14025)]. The approach of the present paper is different from Xiao's one, in as much as Bombieri's method for the study of the bicanonical map [see \textit{E. Bombieri}, Publ. Math., Inst. Hautes Étud. Sci. 42(1972), 171-219 (1973; Zbl 0259.14005)] is mainly used and a precise description of the possible cases in Xiao's list is given.
scientific article; zbMATH DE number 1028145