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

Manuel Blickle, Karl Schwede, Kevin Tucker

Number 24
Author Prof. Dr. Manuel Blickle
Year 2011

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.

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