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

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

2011-11-11

arxiv.org/abs/1111.2762v1

Mathematics

Commutative Algebra

16 pages, comments welcome

Scientific paper

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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