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Surfaces fibrées en courbes de genre deux. (French) Zbl 0579.14028

Lecture Notes in Mathematics. 1137. Berlin etc.: Springer-Verlag. IX, 103 p. DM 21.50 (1985).
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 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.
Reviewer: F.L.Zak

MSC:

14J10 Families, moduli, classification: algebraic theory
14D20 Algebraic moduli problems, moduli of vector bundles
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0349.14021
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