On the behavior of singularities at the \(F\)-pure threshold. With an appendix by Alessandro De Stefani, Jack Jeffries, Zhibek Kadyrsizova, Robert Walker, and George Whelan. (Q2408441)

For a nonzero element \(f \in R\) in a Noetherian ring of characteristic \(p > 0\), the \(F\)-pure threshold \(\mathrm{fpt}(f)\) is the largest positive real number \(t\) for which the pair \((R, f^t)\) is \(F\)-pure. This is in analogy with the log canonical threshold, which is usually considered only in characteristic zero (although this assumption is not necessary). If \(R\) is a polynomial ring over \(\mathbb Q\) and \(f_p\) denotes the reduction of an element \(f \in R\) modulo \(p\), then one has the following chain of inequalities (strict in general): \[ \mathrm{fpt}(f_p) \leq \mathrm{lct}(f_p) \leq \mathrm{lct}(f). \] Based on examples and special cases (e.g. homogeneous polynomials with an isolated singularity), it has been asked whether \(\mathrm{fpt}(f_p) = \mathrm{lct}(f_p)\) whenever \(p\) does not divide the denominator of \(\mathrm{fpt}(f_p)\). (It is known that for \(R\) regular, the latter number is always rational). There is one counterexample in the literature, due to Mustaţă, Takagi and Watanabe [\textit{M. Mustaţă} et al., in: Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27--July 2, 2004. Zürich: European Mathematical Society (EMS). 341--364 (2005; Zbl 1092.32014)], but the authors claim it is ``not as widely known as it should be''. Theorem A provides a new example of a polynomial \(f\) such that \(\mathrm{fpt}(f_p) \lneq \mathrm{lct}(f_p)\), yet \(p\) does not divide the denominator of \(\mathrm{fpt}(f_p)\), for infinitely many primes \(p\). The example is \[ f = x_1^d + \cdots + x_n^d + (x_1 \cdots x_n)^{d-2}, \] for suitable values of \(n\) and \(d\). Call the condition that \(p\) not divide the denominator of \(\mathrm{fpt}(f_p)\) Condition (a), and say that Condition (b) is satisfied if \(\mathrm{fpt}(f_p) = \mathrm{lct}(f_p)\). Then Theorem A says that Condition (a) does not imply Condition (b). In Theorem B, the authors study the consequences of (a) and (b) for the \(F\)-signature function. They prove (under some additional technical conditions) that if (a) holds or ((b) and resolution of singularities) hold, then the left derivative of the \(F\)-signature function of the pair \((R, f)\) at \(\mathrm{fpt}(f)\) is zero. The paper also contains an appendix by a disjoint set of authors (De Stefani, Jeffries, Kadyrsizova, Walker, and Whelan). In the appendix, a series of examples is given where Condition (b) is satisfied, but the test ideal \(\tau \big( R, f^{\mathrm{fpt}(f)} \big)\) is not radical. In particular, this shows that Condition (b) does not imply Condition (a).