Log terminal singularities

Abstract

In this paper we give a new point of view for optimizing the definitions related to the study of singularities of normal varieties, introduced in [dFH09] and further studied in [Urb12a] and [Urb12b], in relation to the Minimal Model Program. We introduce a notion of discrepancy for normal varieties, and we define log terminal + singularities. We use finite generation to relate these new singularities with log terminal singularities (in the sense of [dFH09]).

Some Properties and Examples of Log Terminal+Singularities

“…We have the following generalization of [dFH09], 5.4. In [CU12] this results is used to prove the following proposition.…”

“…The main point is that these pathologies occur even for singularities that are very well behaved. Moreover, as it is argued in [CU12], lt + singularities seems to be the largest class of singularities, at the moment, where it is possible to run a "non-Q-Gorenstein MMP without boundaries". Therefore, the examples presented here are to be considered as a cautioning collection for everyone interested in the project.…”

“…Lemma 2.25([dFH09], 5.4,[CU12], 3.6). For each proper birationational morphism f : Y → X between normal varieties, any m ≥ 2 and any divisor D on X, there exists a weak m-compatible D-boundary with respect to f .…”

“…In [Urb12b], the author proceed with his study of non-Q-Gorenstein canonical singularities, and their relation with divisorial models. In [CU12], the author of this paper and Urbinati introduce a new class of singularities, called log terminal + , or simply lt + , which they prove is rather well behaved. For example, the canonical algebra R(X, K X ) is finitely generated in this case ( [CU12], 5.10, theorem 2.21).…”

Ample Weil divisors

“…In [CU12, 5.2], the notion of log terminal + singularities was introduced: if X is a normal variety over C, X has lt + singularities if for one (equivalently all) log resolution f : Y → X, K Y + f * (−K X ) has coefficients strictly bigger than −1 (for any prime component of the exceptional divisor). By [CU12,5.15], if X has lt + singularities, than is has klt singularities (in the sense of the above theorem) if and only if R(X, −K X ) is finitely generated. Thus the above theorem is true under the hypothesis that X has only lt + singularities.…”