Complex varieties of general type whose canonical systems are composed with pencils. (English) Zbl 0927.14003

Chen, Meng

J. Math. Soc. Japan 51, No. 2, 331-335 (1999).

This paper aims to study a variety of general type whose canonical system is composed with a pencil. This kind of variety admits a natural fibration onto a nonsingular curve. A natural problem is whether the geometric genus of the general fibre of this fibration is bounded. A simple classification is given in this paper. When the object is a nonsingular minimal 3-fold of general type, if the canonical system is composed of an irrational pencil, then the geometric genus of the general fibre is bounded. Otherwise, it seems that the geometric genus of the general fibre is not bounded though no counterexamples have been found.

MSC:

14C20	Divisors, linear systems, invertible sheaves
14D06	Fibrations, degenerations in algebraic geometry
14J10	Families, moduli, classification: algebraic theory
14J29	Surfaces of general type
14J30	\(3\)-folds

Keywords:

variety of general type; canonical system; fibration; minimal 3-fold of general type; bounded geometric genus