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Explicit birational geometry of 3-folds and 4-folds of general type, III

Part of: Birational geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  30 December 2014

Jungkai A. Chen  and
Meng Chen
Jungkai A. Chen
Affiliation:
National Center for Theoretical Sciences, Taipei Office, and Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan email jkchen@ntu.edu.tw
Meng Chen
Affiliation:
Institute of Mathematics & LMNS, Fudan University, Shanghai 200433, PR China email mchen@fudan.edu.cn
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Abstract

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the pluricanonical section index${\it\delta}(V):=\text{min}\{m\mid P_{m}\geqslant 2\}$ since $1\leqslant {\it\delta}(V)\leqslant 18$ due to our previous series (I, II). Based on our further classification to 3-folds with ${\it\delta}(V)\geqslant 13$ and an intensive geometrical investigation to those with ${\it\delta}(V)\leqslant 12$, we prove that $\text{Vol}(V)\geqslant \frac{1}{1680}$ and that the pluricanonical map ${\rm\Phi}_{m}$ is birational for all $m\geqslant 61$, which greatly improves known results. An optimal birationality of ${\rm\Phi}_{m}$ for the case ${\it\delta}(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_{g}\geqslant 2$ in the last section.

Keywords

pluricanonical mapsalgebraic 3-folds and 4-foldslinear systemalgebraic fibrations

MSC classification

Primary: 14E05: Rational and birational maps
Secondary: 14J30: $3$-folds 14J35: $4$-folds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 6 , June 2015 , pp. 1041 - 1082
Copyright
© The Authors 2014 

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