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The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambros paper hold true. The proof is Kawamatas X-method with the aid of the theory of basic slc-trivial fibrations. For the readers convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.
We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wisniewski. Moreover, we establish a generalization for quasi-log canonical pairs.
In this paper, we show that Fujitas basepoint-freeness conjecture for projective quasi-log canonical singularities holds true in dimension three. Immediately, we prove Fujita-type basepoint-freeness for projective semi-log canonical threefolds.
We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connec
An association scheme is called quasi-thin if the valency of each its basic relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of it forms a Klein group with respect to the relation product. It is proved that any Kleinian sch
Let (S, BS) be the log-pair associated with a compactification of a given smooth quasi-projective surface V . Under the assumption that the boundary BS is irreducible, we propose an algorithm, in the spirit of the (log) Sarkisov program, to factorize