F -signature of pairs: continuity, p -fractals and minimal log discrepancies

Abstract

This paper contains a number of observations on the F-signature of triples (R, Δ, 𝔞t) introduced in our previous joint work [M. Blickle, K. Schwede and K. Tucker, ‘F-signature of pairs and the asymptotic behavior of Frobenius splittings’, Preprint, 2011, arXiv:1107.1082]. We first show that the F-signature s(R, Δ, 𝔞t) is continuous as a function of t, and for principal ideals 𝔞 even convex. We then further deduce that, for fixed t, the F-signature is lower semi-continuous as a function on Spec R when R is regular and 𝔞 is principal. We also point out the close relationship of the signature function in this setting to the studies of Monsky and Teixeira on Hilbert–Kunz multiplicity and p-fractals [P. Monsky and P. Teixeira, ‘p-fractals and power series. I. Some 2 variable results’, J. Algebra 280 (2004) 505–536; P. Monsky and P. Teixeira, ‘p-fractals and power series. II. Some applications to Hilbert–Kunz theory’, J. Algebra 304 (2006) 237–255]. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple (R, Δ, 𝔞t) is an upper bound for the F-signature.