Characterization of the 4-canonical birationality of algebraic threefolds

Details Characterization of the 4-canonical birationality of algebraic threefolds Characterization of the 4-canonical birationality of algebraic threefolds

2007-03-20

arxiv.org/abs/math/0703593v1

Mathematics

Algebraic Geometry

25 pages, to appear in Mathematische Zeitschrift

Scientific paper

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities and the geometric genus $p_g(X)\ge 5$. We show that the 4-canonical map $\phi_4$ is {\it not} birational onto its image if and only if $X$ is birationally fibred by a family $\mathscr{C}$ of irreducible curves of geometric genus 2 with $K_X\cdot C_0=1$ where $C_0$ is a general irreducible member in $\mathscr{C}$.

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