Surfaces fibrées en courbes de genre deux
The book under review is devoted to a detailed study of surfaces representable in the form of a family of curves of genus 2 (with possible degenerations). The classification of surfaces with a pencil of rational or elliptic curves is classical, and the classification of surfaces with a pencil of curves of genus 2 (which is the simplest case of surfaces of general type) was begun by \textit{E. Horikawa} [cf. Complex Anal. Algeb. Geom., collect. Pap. dedic. K. Kodaira, 79-90 (1977; Zbl 0349.14021)] who considered them as double coverings of ruled surfaces. The author introduces numerical invariants of surfaces fibered into curves of genus 2 and examines the question of existence of surfaces with given numerical invariants. It is shown that a surface fibered into curves of genus 2 usually has general type; a list of all possible exceptions from this rule is given in section 4. For surfaces of general type there usually exists at most one pencil of curves of genus 2; the author gives a list of all possible exceptions which includes a surface with \(1+20\cdot 19/2=191\) pencils. Finally, the author studies canonical and bicanonical maps for surfaces with a pencil of curves of genus 2.
classification of surfaces with a pencil of curves of genus 2
scientific article; zbMATH DE number 3927090