Frobenius powers
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Mathematische Zeitschrift
Frobenius powers Daniel J. Hernández1 · Pedro Teixeira2 · Emily E. Witt1 Received: 19 September 2018 / Accepted: 27 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.
1 Introduction This article concerns the singularities of algebraic varieties, especially the relationship between the singularities of hypersurfaces and those of more general varieties. Though our main interest is in the prime characteristic setting, we are motivated by the following wellknown result from birational algebraic geometry over the complex numbers: Let a be an ideal of a polynomial ring over C. If f ∈ a is a general C-linear combination of some fixed generators of a, then J ( f t ) = J (at )
for each parameter t in the open unit interval [17, Proposition 9.2.28]. This equality of multiplier ideals immediately implies that the log canonical threshold of such an f ∈ a equals the minimum of 1 and the log canonical threshold lct(a) of a. An important special case is when a is the term ideal of f . In this case, the condition that f is general can be expressed concretely: it suffices to take f nondegenerate with respect to the Newton polyhedron of a [16, Corollary 13]. Consequently, the multiplier ideals and log
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Emily E. Witt [email protected]
1
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
2
Department of Mathematics, Knox College, Galesburg, IL 61401, USA
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canonical threshold of a general polynomial can be computed combinatorially from its term ideal [15]. At present, it is understood that there is an intimate relationship between birational algebraic geometry over fields of characteristic zero and the study of singularities in prime characteristic from the point of view of the Frobenius endomorphism. Therefore, it is natural to ask whether those results relating the singularities of a generic element f ∈ a to those of a extend to the positive characteristic setting, after replacing multiplier ideals and the log canonical threshold with their analogs, namely test ideals and the F-pure threshold. Unfortunately, several obstructions are encount
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