Pluricanonical maps for threefolds of general type

Abstract

In this paper we will prove that for a threefold of general type and large volume the second plurigenera is positive and the fifth canonical map is birational. * The author would like to thank Professor Christopher Hacon for suggesting the problem and many useful conversations and suggestions.

Complex projective 3-fold with non-negative canonical Euler-Poincare characteristic

“…• ϕ 5 := Φ |5KX | is birational if either X is Gorenstein (by Chen, Chen and Zhang [4]) or p g (X) ≥ 4 (by Chen [8]) or K 3 X 0 (by Todorov [30]). r 3 = 5 is optimal.…”

On the geography of threefolds of general type

“…We expect that this is far from optimal and so we make no effort to determine it explicitly. We remark that if Vol( X) 0, then using the results of [5], one can recover c = 2502.…”

Characterization of the 4-canonical birationality of algebraic threefolds

“…Tsuji [31] has ever proved r 3 ≤ 18(2 9 · 3 7 )!. If one requires in addition that either the invariants of X are big (e.g., for p g (X ) ≥ 4 see [6]; for K 3 X 0 see [30]) or X is Gorenstein (see [11]), one may take r 3 = 5. We remark that r 3 can not be 4 because the 4-canonical map of the product of a curve and a surface of type (K 2 , p g ) = (1, 2) is not birational.…”