2000/04/30 by Meng Chen, MENG CHEN
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Applied mathematics #Computer science #Geometry and complex manifolds #Index (typography) #Mathematical analysis #Mathematics #Pure mathematics #Singularity #Stability (learning theory) #math.AG
paper · pdf · doi:10.1017/s030500410100531x
published as Math. Proc. Cambridge Philos. Soc. 131(2001), no.2, 241-264 · 27 pages, The final version, Accepted for publication in Math. Proc. Camb. Phil. Soc
arxiv created 2000/05/18 · openalex publication_date 2001/09/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕ m the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ] mK X [mid ]. Since, once given such a 3-fold X , ϕ m is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m . This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [ 21 , remark in section 1]. To fix the terminology we say that ϕ m is stably birational if ϕ t is birational onto its image for all t [ges ] m . It is well known that the parallel problem in the surface case was solved by Bombieri [ 1 ] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [ 7 ], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X . Denote by r the canonical index of minimal models of X . When r = 1 we know that ϕ 6 is stably birational by virtue of [ 3 , 6 , 13 and 14 ]. When r [ges ] 2, Hanamura proved the following theorem.