Canonical Stability in Terms of Singularity Index for Algebraic Threefolds

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 φm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system |mKX |. Since, once given such a 3-fold X , φm is birational whenever m ≫ 0, thus a quite interesting thing is to find 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 §1). To fix the terminology, we say that φm is stably birational if φt is birational onto its image for all t ≥ m. It is well-known that the parallel problem in surface case was solved by Bombieri ([1]) and others. In 3-dimensional case, many authors have ever studied in quite different ways. Because, in this paper, we are interested in the results obtained by M. Hanamura ([7]), we don’t 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 φ6 is stably birational by virtue of [3], [6], [13] and [14]. When r ≥ 2, M. Hanamura proved the following theorem.