  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]).
