\(F\)-signature of pairs: continuity, \(p\)-fractals and minimal log discrepancies (Q2840167)

Let \(R\) be a \(d\)-dimensional, \(F\)-finite local normal domain. Let \(\Delta\) be an effective \(\mathbb{R}\)-divisor on \(\text{Spec}(R)\), \(\mathfrak{a} \subseteq R\) a nonzero ideal, and \(t\) a nonnegative real number. The paper under review studies the \(F\)-signature of the triple \((R, \Delta, \mathfrak{a}^t)\), denoted by \(s(R, \Delta, \mathfrak{a}^t)\). The authors have defined this number in previous work [Adv. Math. 231, No. 6, 3232--3258 (2012; Zbl 1258.13008)] and this paper investigates its properties in detail. The authors consider the function \(t \to s(R, \Delta, \mathfrak{a}^t)\) and prove that it is a continuous, Lipschitz function, which moreover is convex on \([0, \infty)\), when \(\mathfrak{a}\) is principal. One important part of the paper is dedicated to the case when \(R\) is regular, \(\Delta=0\), and \(\mathfrak{a}=(f)\), \(f \in R\) nonzero. The authors note that the function \(\frac{a}{p^c} \mapsto s(R, f^{a/p^c})\), with \(a, c \in \mathbb{N}\), is a \(p\)-fractal in the sense of \textit{P. Monsky} and \textit{P. Teixeira} [J. Algebra 280, No. 2, 505--536 (2004; Zbl 1082.13016)]. They prove that the left derivative of the function at \(t=1\) equals the \(-s(R/(f))\), where \(s(R)\) is the \(F\)-signature of \(R\), and the right derivative at \(0\) equals \(-\text{e}_{HK}(R/(f))\), where \(\text{e}_{HK}(R)\) is the Hilbert-Kunz multiplicity of \(R\). For fixed real positive \(t\), they show that the functionNEWLINENEWLINENEWLINE\[NEWLINEq \in \text{Spec(R)} \mapsto s(R_q, f^t)NEWLINE\]NEWLINE is lower semi-continuous on \(\text{Spec}(R)\). Finally, a connection to minimal log discrepancies is developed in the last section of the paper.