F-signature function of quotient singularities

Abstract

We study the shape of the F-signature function of a d-dimensional quotient singularity k x 1 , . . . , x d G , and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group G ⊆ Gl(d, k). When G is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples.

Differential symmetric signature in high dimension

“…The proof of Theorem 2.8 shows that the function N → N q=0 frk R Sym q R (Ω R/k ) * * is quasi-polynomial. A similar behaviour for the F-signature function of quotient singularities is observed by the second author and De Stefani in [CDS18].…”

The Shape of Hilbert-Kunz Functions

“…It is also known, from the work of Huneke and Leuschke [58], Singh [98], Yao [119], and Aberbach [1], that the theory of F -signature is closely related to that of Hilbert-Kunz multiplicity . In particular, it is proved by Tucker, Polstra, Caminata and De Stefani [111,83,21] that F -signature functions have the same approximate functional forms as Hilbert-Kunz functions. In the case of affine semigroup rings, the work of Watanabe [113], Bruns [16], Singh [98], and Von Korff [112] show that the same effective methods can be used to compute these two sets of functions and multiplicities.…”

Equimultiplicity in Hilbert–Kunz theory

“…This note grew from the lectures that I delivered at the ICTP school, but the content has changed a lot. Since there are several good introductions to F-invariants (such as [29,8,20]), I decided to omit the proofs of the basic results and instead to focus on the notion of equimultiplicity and its role in the theory of Hilbert-Kunz multiplicity and F-signature. My motivation was the lack of a good introduction to equimultiplicity in general and the feeling that there is still a lot to be done in the study of equimultiplicity for F-invariants, i.e., invariants derived from the Frobenius.…”

“…The purpose of this section is to discuss the basics of the positive characteristic commutative algebra and introduce Hilbert-Kunz multiplicity and F-signature. The proofs and more comprehensive treatments can be found in [20,29,8].…”