Some birationality criteria on 3-folds with p g > 1

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].…”

“…Now take p 563/20 and then take a sufficiently small > 0 in (5) and (6), we have M 2 > 8 and M.C 4 for every curve C passing through a very general point of F. Apply Lemma 3·2, we see that the linear system |K F + M | gives a birational map on F. Since we have seen in (4) that…”

On projective threefolds of general type with small positive geometric genus

“…is nef and big. Since (Q −2 • C 5 ) ≥7 3 > 2, we see that |K F + Q −2 + C 5 || C5 gives a birational map. Thus ϕ 16,X is birational by Lemma 3.1 and Lemma 3.2.(2-i-3).…”

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

“…For any irreducible and reduced curveĈ ⊂ S, passing through a very general point P ∈ S, it is sufficient to prove (σ * (K S 0 ) ·Ĉ) ≥ 2 and so that (L ·Ĉ) > 6. Then the statement follows from either [7,Lemma 2.6] or [16,Proposition 4]. In fact, (σ * (K S 0 ) ·Ĉ) = (K S 0 · C) where C := σ * (Ĉ).…”