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

2011/11/30 by Manuel Blickle, Karl Schwede, Kevin Tucker · 2 citations
Mathematics · #math.AC #math.AG #msc:13A35 #msc:13D40 #msc:14B05 #msc:13H10 #msc:14F18

paper · pdf · doi:10.1112/jlms/jds070

published as J. London Math. Soc. vol. 87, no. 3, 802--818, (2013) · 17 pages, exposition improved, typos corrected, to appear in Journal of the London Mathematical Society

arxiv created 2012/09/07 · arxiv updated 2013/07/16

Abstract

This paper contains a number of observations on the F-signature of triples (R,Δ,\ba^t) introduced in our previous joint work. We first show that the F-signature s(R,Δ,\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,Δ,\ba^t) is an upper bound for the F-signature.

Cited by

• Limit F -signature functions of two-variable binomial hypersurfaces
• Limit F-signature functions of two-variable binomial hypersurfaces

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