Skip to main content Accessibility help
×
Home

THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS

  • TAKUMI MURAYAMA (a1)

Abstract

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$ -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$ -regular varieties over arbitrary fields.

Copyright

Footnotes

Hide All

This material is based upon work supported by the National Science Foundation under grant nos. DMS-1501461 and DMS-1701622.

Footnotes

References

Hide All
[1] Angehrn, U. and Siu, Y. T., Effective freeness and point separation for adjoint bundles , Invent. Math. 122 (1995), 291308.
[2] Avramov, L. L., Flat morphisms of complete intersections , Soviet Math. Dokl. 16 (1975), 14131417; Translated from the Russian by D. L. Johnson.
[3] Bingener, J. and Flenner, H., “ On the fibers of analytic mappings ”, in Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 45101.
[4] Birkar, C., The augmented base locus of real divisors over arbitrary fields , Math. Ann. 368 (2017), 905921.
[5] Boucksom, S., Broustet, A. and Pacienza, G., Uniruledness of stable base loci of adjoint linear systems via Mori theory , Math. Z. 275 (2013), 499507.
[6] Burgos Gil, J. I., Gubler, W., Jell, P., Künnemann, K. and Martin, F., Differentiability of non-archimedean volumes and non-archimedean Monge–Ampère equations , Algebr. Geom. (2019), to appear, With an appendix by R. Lazarsfeld, arXiv:1608.01919v6 [math.AG].
[7] Cacciola, S. and Di Biagio, L., Asymptotic base loci on singular varieties , Math. Z. 275 (2013), 151166.
[8] Cutkosky, S. D., Teissier’s problem on inequalities of nef divisors , J. Algebra Appl. 14 (2015), 1540002, 37 pp.
[9] Datta, R. and Murayama, T., Permanence properties of $F$ -injectivity, preprint, 2019. arXiv:1906.11399v1 [math.AC].
[10] Datta, R. and Smith, K. E., Frobenius and valuation rings , Algebra Number Theory 10 (2016), 10571090; See also [ 11 ].
[11] Datta, R. and Smith, K. E., Correction to the article “Frobenius and valuation rings” , Algebra Number Theory 11 (2017), 10031007.
[12] de Fernex, T., Küronya, A. and Lazarsfeld, R., Higher cohomology of divisors on a projective variety , Math. Ann. 337 (2007), 443455.
[13] de Fernex, T. and Mustaţă, M., Limits of log canonical thresholds , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 491515.
[14] Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M., Asymptotic invariants of line bundles , Pure Appl. Math. Q. 1 (2005), 379403.
[15] Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M., Asymptotic invariants of base loci , Ann. Inst. Fourier (Grenoble) 56 (2006), 17011734.
[16] Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M., Restricted volumes and base loci of linear series , Amer. J. Math. 131 (2009), 607651.
[17] Enescu, F. and Hochster, M., The Frobenius structure of local cohomology , Algebra Number Theory 2 (2008), 721754.
[18] Fedder, R., F-purity and rational singularity , Trans. Amer. Math. Soc. 278 (1983), 461480.
[19] Fedder, R. and Watanabe, K.-i., “ A characterization of F-regularity in terms of F-purity ”, in Commutative Algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 15 , Springer, New York, 1989, 227245.
[20] Flenner, H., O’Carroll, L. and Vogel, W., Joins and Intersections, Springer Monogr. Math., Springer, Berlin, 1999.
[21] Fulger, M., Kollár, J. and Lehmann, B., Volume and Hilbert function of ℝ-divisors , Michigan Math. J. 65 (2016), 371387.
[22] Gabber, O., “ Notes on some t-structures ”, in Geometric Aspects of Dwork Theory. Vol. II, Walter de Gruyter, Berlin, 2004, 711734.
[23] Grothendieck, A., Séminaire de géométrie algébrique du Bois Marie, 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Doc. Math. (Paris), 4 , Soc. Math. France, Paris, 2005, With an exposé by Mme M. Raynaud, With a preface and edited by Y. Laszlo, Revised reprint of the 1968 French original.
[24] Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II , Publ. Math. Inst. Hautes Études Sci. 24 (1965), 1231.
[25] Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III , Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1255.
[26] Hara, N. and Takagi, S., On a generalization of test ideals , Nagoya Math. J. 175 (2004), 5974.
[27] Hara, N. and Yoshida, K., A generalization of tight closure and multiplier ideals , Trans. Amer. Math. Soc. 355 (2003), 31433174.
[28] Hartshorne, R., Residues and Duality, Lecture Notes in Mathematics, 20 , Springer, Berlin–New York, 1966, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, With an appendix by P. Deligne.
[29] Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, 52 , Springer, New York–Heidelberg, 1977.
[30] Hashimoto, M., F-pure homomorphisms, strong F-regularity, and F-injectivity , Comm. Algebra 38 (2010), 45694596.
[31] Hashimoto, M., “ F-purity of homomorphisms, strong F-regularity, and F-injectivity ”, in The 31st Symposium on Commutative Algebra in Japan (Osaka, 2009), 2010, 916. http://www.math.okayama-u.ac.jp/∼hashimoto/paper/Comm-Alg/31-all.pdf.
[32] Hauser, H., On the problem of resolution of singularities in positive characteristic (or: a proof we are still waiting for) , Bull. Amer. Math. Soc. (N.S.) 47 (2010), 130.
[33] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon–Skoda theorem , J. Amer. Math. Soc. 3 (1990), 31116.
[34] Hochster, M. and Huneke, C., F-regularity, test elements, and smooth base change , Trans. Amer. Math. Soc. 346 (1994), 162.
[35] Hochster, M. and Roberts, J. L., Rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay , Adv. Math. 13 (1974), 115175.
[36] Hochster, M. and Roberts, J. L., The purity of the Frobenius and local cohomology , Adv. Math. 21 (1976), 117172.
[37] Ito, A., Okounkov bodies and Seshadri constants , Adv. Math. 241 (2013), 246262.
[38] de Jong, A. J., Smoothness, semi-stability and alterations , Publ. Math. Inst. Hautes Études Sci. 83 (1996), 5193.
[39] Jonsson, M. and Mustaţă, M., Valuations and asymptotic invariants for sequences of ideals , Ann. Inst. Fourier (Grenoble) 62 (2012), 21452209.
[40] Kleiman, S. L., “ The Picard scheme ”, in Fundamental Algebraic Geometry, Math. Surveys Monographs, 123 , American Mathematical Society, Providence, RI, 2005, 235321.
[41] Küronya, A., Asymptotic cohomological functions on projective varieties , Amer. J. Math. 128 (2006), 14751519.
[42] Kunz, E., On Noetherian rings of characteristic p , Amer. J. Math. 98 (1976), 9991013.
[43] Lazarsfeld, R., Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, Ergeb. Math. Grenzgeb. (3), 48 , Springer, Berlin, 2004.
[44] Lazarsfeld, R., Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Ergeb. Math. Grenzgeb. (3), 49 , Springer, Berlin, 2004.
[45] Ma, L., Finiteness properties of local cohomology for F-pure local rings , Int. Math. Res. Not. IMRN 20 (2014), 54895509.
[46] Ma, L., Polstra, T., Schwede, K. and Tucker, K., F-signature under birational morphisms , Forum Math. Sigma 7 (2019), e11, 20 pp.
[47] Manaresi, M., Some properties of weakly normal varieties , Nagoya Math. J. 77 (1980), 6174.
[48] Matsumura, H., Commutative Ring Theory, 2nd ed., Cambridge Stud. Adv. Math., 8 , Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid.
[49] Murayama, T., Frobenius–Seshadri constants and characterizations of projective space , Math. Res. Lett. 25 (2018), 905936.
[50] Murayama, T., Seshadri constants and Fujita’s conjecture via positive characteristic methods, Ph.D. thesis, University of Michigan, Ann Arbor, 2019, 206 pp.
[51] Mustaţă, M., “ The non-nef locus in positive characteristic ”, in A Celebration of Algebraic Geometry, Clay Math. Proc., 18 , American Mathematical Society, Providence, RI, 2013, 535551.
[52] Nakayama, N., Zariski-decomposition and abundance, MSJ Mem. 14 , Math. Soc. Japan, Tokyo, 2004.
[53] Nayak, S., Compactification for essentially finite-type maps , Adv. Math. 222 (2009), 527546.
[54] Patakfalvi, Zs., Schwede, K. and Tucker, K., “ Positive characteristic algebraic geometry ”, in Surveys on Recent Developments in Algebraic Geometry, Proc. Sympos. Pure Math., 95 , American Mathematical Society, Providence, RI, 2017, 3380.
[55] Quy, P. H. and Shimomoto, K., F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0 , Adv. Math. 313 (2017), 127166.
[56] Raynaud, M., “ Contre-exemple au ‘vanishing theorem’ en caractéristique p > 0 ”, in C. P. Ramanujam—A Tribute, Tata Inst. Fund. Res. Studies in Math., 8 , Springer, Berlin–New York, 1978, 273278.
[57] Sato, K., Stability of test ideals of divisors with small multiplicity , Math. Z. 288 (2018), 783802.
[58] Schwede, K., F-injective singularities are Du Bois , Amer. J. Math. 131 (2009), 445473.
[59] Schwede, K., Centers of F-purity , Math. Z. 265 (2010), 687714.
[60] Schwede, K., Test ideals in non-ℚ-Gorenstein rings , Trans. Amer. Math. Soc. 363 (2011), 59255941.
[61] Schwede, K. and Tucker, K., “ A survey of test ideals ”, in Progress in Commutative Algebra 2, Walter de Gruyter, Berlin, 2012, 3999.
[62] Takagi, S. and Watanabe, K.-i., F-singularities: applications of characteristic p methods to singularity theory , Sugaku Expositions 31 (2018), 142; Translated from the Japanese by the authors.
[63] Tanaka, H., Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field , Internat. J. Math. 28 (2017), 1750030, 13 pp.
[64] Tanaka, H., Minimal model program for excellent surfaces , Ann. Inst. Fourier (Grenoble) 68 (2018), 345376.
[65] Tanaka, H., Abundance theorem for surfaces over imperfect fields , Math. Z. (2019), to appear arXiv:1502.01383v5 [math.AG], doi:10.1007/s00209-019-02345-2.
[66] Vélez, J. D., Openness of the F-rational locus and smooth base change , J. Algebra 172 (1995), 425453.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed