On birational geometry of minimal threefolds with numerically trivial canonical divisors

Abstract

Abstract. For a minimal 3-fold X with KX ≡ 0 and a nef and big Weil divisor L on X, we investigate the birational geometry inspired by L. We prove that |mL| and |KX + mL| give birational maps for all m ≥ 17. The result remains true under weaker assumption that L is big and has no stable base components.

Note on quasi-polarized canonical Calabi–Yau threefolds

“…These two questions have already been investigated by several mathematicians in various different settings [6,13,14] etc. Our first result in this note can be viewed as a generalization of [ The estimate is sharp as showed by a general weighted hypersurface of degree 10 in the weighted projective space P(1, 1, 1, 2, 5).…”

“…Our first result in this note can be viewed as a generalization of [ The estimate is sharp as showed by a general weighted hypersurface of degree 10 in the weighted projective space P(1, 1, 1, 2, 5). We remark also that we have always h 0 (X , L) ≥ 1 by [8,Proposition 4.1] and the morphism Φ |5L| is always birational onto its image by [6,Theorem 1.7]. The basepoint freeness of |4H | is an easy consequence of [12,Theorem 24] and the existence of semi-log canonical member in |H | (cf.…”

“…On the other hand, we have always h 0 (X , H ) ≥ n − 2 in Theorem 4 (cf. [11,Theorem 1.2]), and if X is a general weighted complete intersection of type (6,6) in the weighted projective space P(1, . .…”

“…, 1, 2, 2, 3, 3). However, if X is a weighted complete intersection of type (6,6) in the weighted projective space P(1, . .…”

On irregular threefolds and fourfolds with numerically trivial canonical bundle

“…For dim X ≤ 4, we have the following known results: Theorem 1.1 (cf. [21], [8], [16], [9]). Let X be a smooth projective variety with K X ≡ 0, L a nef and big divisor, and T a divisor such that T ≡ 0.…”