We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacons inversion of adjunction for log canonical centers of arbitrary codimension.
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 speakin
g, 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 adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.
The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable
assumptions. It recovers Kawakitas inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As applications, we study lengths of extremal rational curves and so on.
We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers
Svaldis question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.
We show that the Nakai--Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai--Moishezon ampleness criterion for real line bundles and the Nakai--Moishezon ampleness criterion for
real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkars characterization of augmented base loci of real divisors on projective schemes.
The main purpose of this paper is to make Nakayamas theorem more accessible. We give a proof of Nakayamas theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayamas theorem on algebrai
c varieties over any algebraically closed field of arbitrary characteristic although Nakayamas original statement is formulated for complex analytic spaces.
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
ted. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
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.
