Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds

Details Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds

2002-06-10

arxiv.org/abs/math/0206097v4

Mathematics

Algebraic Geometry

AMS-Latex, 9 pages, Proc. AMS (to appear)

Scientific paper

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Gorenstein object here. It is open whether the degree of the canonical map is universally bounded when the canonical map is generically finite.

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