On an efficient induction step with Nklt(X,D) — notes to Todorov

Abstract

Applying the effective induction on Nklt(X, D) developed by Hacon-McKernan and Takayama, Todorov proved that ϕ 5 is birational for projective 3-folds V with vol(V ) ≫ 0, which was recently improved by Di Biagio in loosing the volume constraint. The observation is that the least efficient induction step can be studied in an alternative way, which allows us to assert Todorov's statement for vol(V ) > 12 3 . The 4-dimensional analog is also given in this note. The idea works well for all dimensions.

The third and fourth pluricanonical maps of threefolds of general type

“…We remark that the fourth pluricanonical map for threefolds of general type was first studied by M. Chen and D.-Q. Combined with techniques developed in [4], [5] and [9], we are able to further improve the results of [11]. Combined with techniques developed in [4], [5] and [9], we are able to further improve the results of [11].…”

“…As in [4] (or see [11] for a more precise treatment), we have two cases in consideration. As in [4] (or see [11] for a more precise treatment), we have two cases in consideration.…”

“…By a result of McKernan [20] (see also [4,11,22]), there is a generically finite dominant morphism ν : X → X and a fibration f : X → B,…”

“…We start with Takayama's induction, and follow closely [4,11,22]. By [21,Theorem 3•1], given any rational number 0 < < 1, there is a birational modification μ : X → X , where X is a nonsingular projective threefold, and a decomposition…”

“…Zhang [10], and they raised the question of characterizing non-birationality of ϕ 4 by (1, 2)-surface fibration. Combined with techniques developed in [4], [5] and [9], we are able to further improve the results of [11].…”