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$F$-SIGNATURE UNDER BIRATIONAL MORPHISMS

Part of: Local theory General commutative ring theory Cycles and subschemes

Published online by Cambridge University Press:  17 April 2019

LINQUAN MA ,
THOMAS POLSTRA ,
KARL SCHWEDE  and
KEVIN TUCKER
LINQUAN MA
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA; ma326@purdue.edu
THOMAS POLSTRA
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA; polstra@math.utah.edu, schwede@math.utah.edu
KARL SCHWEDE
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA; polstra@math.utah.edu, schwede@math.utah.edu
KEVIN TUCKER
Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60607, USA; kftucker@uic.edu

Abstract

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We study $F$-signature under proper birational morphisms $\unicode[STIX]{x1D70B}:Y\rightarrow X$, showing that $F$-signature strictly increases for small morphisms or if $K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.

MSC classification

Primary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 14B05: Singularities 14C20: Divisors, linear systems, invertible sheaves
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e11
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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