F-SINGULARITIES VIA ALTERATIONS Blickle, Manuel; Schwede, Karl; Tucker, Kevin
American journal of mathematics,
02/2015, Letnik:
137, Številka:
1
Journal Article
Recenzirano
Odprti dostop
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.
We prove that the
F
-jumping numbers of the test ideal
are discrete and rational under the assumptions that
X
is a normal and
F
-finite scheme over a field of positive characteristic
p
,
K
X
+ Δ is
...-Cartier of index not divisible
p
, and either
X
is essentially of finite type over a field or the sheaf of ideals
is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.
F-thresholds of hypersurfaces BLICKLE, Manuel; MUSTATA, Mircea; SMITH, Karen E
Transactions of the American Mathematical Society,
12/2009, Letnik:
361, Številka:
12
Journal Article
Recenzirano
Odprti dostop
We use the D-module theoretic description of generalized test ideals to show that in any F-finite regular ring the F-thresholds of hypersurfaces are discrete and rational. Furthermore we show that ...any limit of F-pure thresholds of principal ideals in bounded dimension is again an F-pure threshold; hence in particular the limit is rational.
We generalize F-signature to pairs (R,D) where D is a Cartier subalgebra on R as defined by the first two authors. In particular, we show the existence and positivity of the F-signature for any ...strongly F-regular pair. In one application, we answer an open question of Aberbach and Enescu by showing that the F-splitting ratio of an arbitrary F-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the F-signature and the F-splitting ratio in the spirit of the work of R. Fedder.
Functorial test modules Blickle, Manuel; Stäbler, Axel
Journal of pure and applied algebra,
April 2019, 2019-04-00, Letnik:
223, Številka:
4
Journal Article
Recenzirano
In this article we introduce a slight modification of the definition of test modules which is an additive functor τ on the category of coherent Cartier modules. We show that in many situations this ...modification agrees with the usual definition of test modules. Furthermore, we show that for a smooth morphism f:X→Y of F-finite schemes one has a natural isomorphism f!∘τ≅τ∘f!. If f is quasi-finite and of finite type we construct a natural transformation τ∘f⁎→f⁎∘τ.
A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed p and d, there ...exists a number \epsilon(d,p) > 0 such that for any nonregular unmixed ring R its Hilbert-Kunz multiplicity is at least 1+\epsilon(d,p). We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and F-rational.
Frobenius on the Cohomology of Thickenings Bhatt, Bhargav; Blickle, Manuel; Lyubeznik, Gennady ...
International mathematics research notices,
05/2024, Letnik:
2024, Številka:
10
Journal Article
Recenzirano
Odprti dostop
Abstract We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds—that is, ...independent of the characteristic—on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.
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