The Relative Pluricanonical Stability for 3-folds of General Type

The aim of this paper is to improve a theorem of János Kollár by a different method. For a given smooth Complex projective threefold X of general type, suppose the plurigenus Pk(X) ≥ 2, Kollár proved that the (11k + 5)-canonical map is birational. Here we show that either the (7k+3)-canonical map or the (7k+5)-canonical map is birational and the (13k+6)-canonical map is stably birational onto its image. If Pk(X) ≥ 3, then the m-canonical map is birational for m ≥ 10k + 8. In particular, φ12 is birational when pg(X) ≥ 2 and φ11 is birational when pg(X) ≥ 3. Introduction Let X be a smooth projective 3-fold of general type defined over C and denote by φm the m-canonical map of X , which is the rational map associated with the linear system |mKX |. Let Pk(X) := h (X,OX(kKX)) for any positive integer k, we usually call Pk(X) the k-th plurigenus of X which is a birational invariant. For a given positive integer m0, we say that φm0 is stably birational if φm is birational onto its image for all m ≥ m0. Since the Kodaira dimension kod(X) = 3, φm is birational for m ≫ 0. In this paper, we consider the following Problem. Suppose Pk(X) ≥ 2, for which value m0(k), does |m0(k)KX | define a stably birational map onto its image? In 1986, Kollár ([5, Corollary 4.8]) first gave an effective result and proved that the (11k + 5)-canonical map is birational if Pk(X) ≥ 2. However, his method can not tell whether φm is still birational for all m > 11k + 5. On the other hand, it seems to us that the number 11k + 5 is not the optimal one. This paper aims to present a better result as the following Main Theorem. Let X be a nonsingular projective threefold of general type and suppose Pk(X) ≥ 2, then (i) either φ7k+3 or φ7k+5 is birational onto its image; (ii) φ13k+6 is stably birational onto its image; (iii) φ10k+8 is stably birational providing that Pk(X) ≥ 3. 1991 Mathematics Subject Classification. Primary 14C20, 14E05, 14E35. The author was supported in part by the Abdus Salam International Centre for Theoretical Physics and the National Natural Science Foundation of China