Many results are known about test ideals and F -singularities for Q-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra OX ⊕ OX (−KX) ⊕ OX (−2KX ) ⊕ . . . is finitely generated (or more generally, in the log setting for −KX − ∆). In particular, we show that the F -jumping numbers of τ (X, a t ) are discrete and rational. We show that test ideals τ (X) can be described by alterations as in Blickle-Schwede-Tucker (and hence show that splinters are strongly F -regular in this setting -recovering a result of Singh). We demonstrate that multiplier ideals reduce to test ideals under reduction modulo p when the symbolic Rees algebra is finitely generated. We prove that Hartshorne-Speiser-Lyubeznik-Gabber type stabilization still holds. We also show that test ideals satisfy global generation properties in this setting.
We define and study positivity (nefness, amplitude, bigness and pseudo-effectiveness) for Weil divisors on normal projective varieties. We prove various characterizations, vanishing and non-vanishing theorems for cohomology, global generations statements, and a result related to log Fano.
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]).
In [dFH09], de Fernex and Hacon started the study of singularities on non-Q-Gorenstein varieties using pullbacks of Weil divisors. 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. In this paper we will continue the study of lt + singularities, and we will show that they can be detected by a multiplier ideal, that they satisfy a Bertini type result, inversion of adjunction and small deformation invariance, and that they are naturally related to rational singularities. Finally, we will provide a list of examples (all of them with lt + singularities) of the pathologies that can occur in the study of non-Q-Gorenstein singularities.
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