F -singularities via alterations

Abstract

Abstract. We give characterizations of test ideals and F -rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly, we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further demonstrate how these vanishing theorems may be used to extend sections.

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

“…For example, we further strengthen the association between multiplier and test ideals by generalizing the main results of [BST11] and [STZ12] from the case of a principal ideal to an arbitrary ideal.…”

“…Roughly speaking, the next result does this where π : Y − → X is the normalized blowup of a. Along the way, we make use of the parameter test sheaf τ (ω Y ) [Smi95,BST11] which replaces both D and K Y in the above description. Recall that since Y is normal and ω Y is reflexive, the map Tr : F * ω U − → ω U on the smooth locus U ⊆ Y extends to all of Y .…”

“…, whose existence is guaranteed by [BST11], and then sets η = π • γ. Similarly, after incorporating the scaling coefficient t into the above effective computation result in Theorem 5.1, we will be able to prove Theorem B on the discreteness and rationality of F -jumping numbers by using the above characterization to reduce to the principal case on the normalized blowup.…”

“…Hence we will use the parameter test modules in the sections that follow wherever we believe that it conceptually simplifies certain arguments. This perspective was used previously in [BST11] and [STZ12].…”

“…In light of such difficulties, as with the results from [BST11] and [STZ12] generalized above, a number of prominent results for test ideals have been known previously only in the principal case. Our new perspective allows us to realize many of these statements in full generality, roughly showing that the test ideals of non-principal ideals behave nearly as well as those of principal ideals.…”