We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it and reflects give conditions on when a stable objects, singularity defect categories, respectively. In appendix, we direct proof following known result: for an category with enough projectives injectives, its global coincides injective dimension.

Abstract In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, $${\mathbb {Q}}$$ Q -Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numeric...

Let X be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by F a smooth model of a generic irreducible element in fibers of φ1 and so F is a curve or a smooth surface. The main result is that there is a computable constant K independent of X such that g(F ) ≤ 647 or pg(F ) ≤ 38 whenever pg(X) ≥ K.