Hostname: page-component-84b7d79bbc-5lx2p Total loading time: 0 Render date: 2024-08-02T14:23:53.149Z Has data issue: false hasContentIssue false
  • English
  • Français

$F$-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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Aberbach, I. M. and Enescu, F., ‘The structure of F-pure rings’, Math. Z. 250(4) (2005), 791–806.Google Scholar
Aberbach, I. M. and Leuschke, G. J., ‘The F-signature and strong F-regularity’, Math. Res. Lett. 10(1) (2003), 51–56.Google Scholar
Bhatt, B., Gabber, O. and Olsson, M., ‘Finiteness of étale fundamental groups by reduction modulo  $p$ ’, Preprint, 2017, ArXiv e-prints.Google Scholar
Birkar, C., ‘Existence of flips and minimal models for 3-folds in char p ’, Ann. Sci. Éc. Norm. Supér. (4) 49(1) (2016), 169–212.Google Scholar
Blickle, M. and Schwede, K., ‘ p -1 -linear maps in algebra and geometry’, inCommutative Algebra (Springer, New York, 2013), 123–205.Google Scholar
Blickle, M., Schwede, K. and Tucker, K., ‘ F-signature of pairs and the asymptotic behavior of Frobenius splittings’, Adv. Math. 231(6) (2012), 3232–3258.Google Scholar
Bruns, W. and Vetter, U., Determinantal Rings, Lecture Notes in Mathematics, 1327 (Springer, Berlin, 1988).Google Scholar
Carvajal-Rojas, J., Schwede, K. and Tucker, K., ‘Fundamental groups of F-regular singularities via F-signature’, Ann. Sci. Éc. Norm. Supér. (4) 51(4) (2018), 993–1016.Google Scholar
De Stefani, A., Polstra, T. and Yao, Y., ‘Globalizing $F$ -invariants’, Preprint, 2016, ArXiv e-prints.Google Scholar
Fedder, R., ‘ F-purity and rational singularity’, Trans. Amer. Math. Soc. 278(2) (1983), 461–480.Google Scholar
de Fernex, T., Küronya, A. and Lazarsfeld, R., ‘Higher cohomology of divisors on a projective variety’, Math. Ann. 337(2) (2007), 443–455.Google Scholar
Glassbrenner, D., ‘Strongly F-regularity in images of regular rings’, Proc. Amer. Math. Soc. 124(2) (1996), 345–353.Google Scholar
Hara, N. and Yoshida, K.-I., ‘A generalization of tight closure and multiplier ideals’, Trans. Amer. Math. Soc. 355(8) (2003), 3143–3174 (electronic).Google Scholar
Huneke, C. and Leuschke, G. J., ‘Two theorems about maximal Cohen–Macaulay modules’, Math. Ann. 324(2) (2002), 391–404.Google Scholar
Huneke, C. and Swanson, I., Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, 336 (Cambridge University Press, Cambridge, 2006).Google Scholar
de Jong, A. J., ‘Smoothness, semi-stability and alterations’, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93.Google Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134 (Cambridge University Press, Cambridge, 1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.Google Scholar
Kunz, E., ‘Characterizations of regular local rings for characteristic p ’, Amer. J. Math. 91 (1969), 772–784.Google Scholar
Lazarsfeld, R., ‘Classical setting: line bundles and linear series’, inPositivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48 (Springer, Berlin, 2004).Google Scholar
Lazarsfeld, R. and Mustaţă, M., ‘Convex bodies associated to linear series’, Ann. Sci. Éc. Norm. Supér. (4) 42(5) (2009), 783–835.Google Scholar
Li, C., Liu, Y. and Xu, C., ‘A guided tour to normalized volume’, Preprint, 2018, ArXiv e-prints.Google Scholar
Liu, Y., ‘The $F$ -volume of singularities in positive characteristic’, 2018, in preparation.Google Scholar
Liu, Y. and Xu, C., ‘ $K$ -stability of cubic threefolds’, Preprint, 2017, ArXiv e-prints.Google Scholar
Murayama, T., ‘The gamma construction and asymptotic invariants of line bundles over arbitrary fields’, Preprint, 2018, ArXiv e-prints.Google Scholar
Mustaţă, M., ‘The non-nef locus in positive characteristic’, inA Celebration of Algebraic Geometry, Clay Math. Proc., 18 (American Mathematical Society, Providence, RI, 2013), 535–551.Google Scholar
Polstra, T., ‘Uniform bounds in F-finite rings and lower semi-continuity of the F-signature’, Trans. Amer. Math. Soc. 370(5) (2018), 3147–3169.Google Scholar
Polstra, T. and Tucker, K., ‘ F-signature and Hilbert–Kunz multiplicity: a combined approach and comparison’, Algebra Number Theory 12(1) (2018), 61–97.Google Scholar
Sannai, A., ‘On dual F-signature’, Int. Math. Res. Not. IMRN 2015(1) (2015), 197–211.Google Scholar
Schwede, K. and Smith, K. E., ‘Globally F-regular and log Fano varieties’, Adv. Math. 224(3) (2010), 863–894.Google Scholar
Smith, K. E. and Van den Bergh, M., ‘Simplicity of rings of differential operators in prime characteristic’, Proc. Lond. Math. Soc. (3) 75(1) (1997), 32–62.Google Scholar
Tucker, K., ‘ F-signature exists’, Invent. Math. 190(3) (2012), 743–765.Google Scholar
Xu, C., ‘Finiteness of algebraic fundamental groups’, Compos. Math. 150(3) (2014), 409–414.Google Scholar
Yao, Y., ‘Observations on the F-signature of local rings of characteristic p ’, J. Algebra 299(1) (2006), 198–218.Google Scholar