This paper contains a number of observations on the {F-signature} of
triples (R,\Delta,\ba^t) introduced in our previous joint work. We first show
that the F-signature s(R,\Delta,\ba^t) is continuous as a function of t,
and for principal ideals \ba even convex. We then further deduce, for fixed
t, that the F-signature is lower semi-continuous as a function on \Spec R
when R is regular and \ba is principal. We also point out the close
relationship of the signature function in this setting to the works of Monsky
and Teixeira on Hilbert-Kunz multiplicity and p-fractals. Finally, we
conclude by showing that the minimal log discrepancy of an arbitrary triple
(R,\Delta,\ba^t) is an upper bound for the F-signature.Comment: 17 pages, exposition improved, typos corrected, to appear in Journal
of the London Mathematical Societ
