On the number of compatibly Frobenius split subvarieties, prime $F$-ideals, and log canonical centers

Schwede, Karl; Tucker, Kevin

Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1515-1531.

Soit $X$ une variété projective Frobenius scindée avec un scindage de Frobenius $\theta$. Dans cet article nous donnons une borne optimale et uniforme sur le nombre de sous-variétés de $X$ qui sont compatibles avec le scindage de Frobenius $\theta$. De même, nous donnons une borne sur le nombre de $F$-idéaux d’un anneau local $F$-fini $F$-pur. Enfin, nous donnons également une borne sur le nombre de centres canoniques logarithmiques d’un paire canonique logarithmique. Cette dernière variante étend un cas particulier d’un résultat de Helmke.

Let $X$ be a projective Frobenius split variety with a fixed Frobenius splitting $\theta$. In this paper we give a sharp uniform bound on the number of subvarieties of $X$ which are compatibly Frobenius split with $\theta$. Similarly, we give a bound on the number of prime $F$-ideals of an $F$-finite $F$-pure local ring. Finally, we also give a bound on the number of log canonical centers of a log canonical pair. This final variant extends a special case of a result of Helmke.

Classification : 13A35, 14B05, 14J17
Keywords: Frobenius split, compatibly Frobenius split subvariety, log canonical center, F-ideal
Mot clés : Frobenius scindé, compatiblement Frobenius scindé sous-variété, centres canoniques logarithmiques, F-idéaux

[1] Ambro, F. Basic properties of log canonical centers (arXiv:math.AG/0611205)

[2] Ambro, F. The locus of log canonical singularities (arXiv:math.AG/9806067)

[3] Ambro, F. Quasi-log varieties, Tr. Mat. Inst. Steklova, Volume 240 (2003) no. Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, pp. 220-239 MR1993751 (2004f:14027) | MR | Zbl

[4] Brion, M.; Kumar, S. Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhäuser Boston Inc., Boston, MA, 2005 MR2107324 (2005k:14104) | MR | Zbl

[5] Eisenbud, D. Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 (With a view toward algebraic geometry) | MR | Zbl

[6] Enescu, F.; Hochster, M. The Frobenius structure of local cohomology, Algebra Number Theory, Volume 2 (2008) no. 7, pp. 721-754 (MR2460693) | DOI | MR | Zbl

[7] Ferrand, D. Conducteur, descente et pincement, Bull. Soc. Math. France, Volume 131 (2003) no. 4, pp. 553-585 MR2044495 (2005a:13016) | Numdam | MR | Zbl

[8] Fujino, O. Introduction to the log minimal model program for log canonical pairs (2009, http://www.math.kyoto-u.ac.jp/preprint/2009/03fujino.pdf)

[9] Fujino, O. The indices of log canonical singularities, Amer. J. Math., Volume 123 (2001) no. 2, pp. 229-253 MR1828222 (2002c:14003) | DOI | MR | Zbl

[10] Greco, S.; Traverso, C. On seminormal schemes, Compositio Math., Volume 40 (1980) no. 3, pp. 325-365 MR571055 (81j:14030) | Numdam | MR | Zbl

[11] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965) no. 24, pp. 231 | Numdam | MR | Zbl

[12] Hartshorne, R.; Speiser, R. Local cohomological dimension in characteristic $p$, Ann. of Math. (2), Volume 105 (1977) no. 1, pp. 45-79 MR0441962 (56 #353) | DOI | MR | Zbl

[13] Helmke, S. On Fujita’s conjecture, Duke math. J., Volume 88 (1997) no. 2, pp. 201-216 | DOI | MR | Zbl

[14] Kawamata, Y. On Fujita’s freeness conjecture for $3$-folds and $4$-folds, Math. Ann., Volume 308 (1997) no. 3, pp. 491-505 MR1457742 (99c:14008) | DOI | MR | Zbl

[15] Kollár, J.; 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. MR1658959 (2000b:14018) | MR | Zbl

[16] Kumar, S.; Mehta, V. B. Finiteness of the number of compatibly-split subvarieties (to appear in Int. Math. Res. Not. arXiv:0901.2098) | Zbl

[17] Kunz, E. On Noetherian rings of characteristic $p$, Amer. J. Math., Volume 98 (1976) no. 4, pp. 999-1013 MR0432625 (55 #5612) | DOI | MR | Zbl

[18] Lazarsfeld, R. Positivity 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-Verlag, Berlin, 2004 Classical setting: line bundles and linear series. MR2095471 (2005k:14001a) | MR

[19] Lazarsfeld, R. Positivity in algebraic geometry. II, 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], 49, Springer-Verlag, Berlin, 2004 Positivity for vector bundles, and multiplier ideals. MR2095472 (2005k:14001b) | MR | Zbl

[20] Mehta, V. B.; Ramanathan, A. Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2), Volume 122 (1985) no. 1, pp. 27-40 MR799251 (86k:14038) | DOI | MR | Zbl

[21] Schwede, K. Centers of $F$-purity (to appear in Math. Z. arXiv:0807.1654)

[22] Schwede, K. $F$-adjunction (to appear in Algebra Number Theory. arXiv:0901.1154)

[23] Schwede, K. Gluing schemes and a scheme without closed points, Proceedings of the 2002 John H. Barrett Memorial Lectures Conference on Algebraic and Arithmetic Geometry (P. T. Y. Kachi, S. Mulay, ed.), Contemporary Mathematics, 386, American Mathematical Society, Providence, RI, 2005 (p. 157–172) | MR

[24] Schwede, K.; Smith, K. Globally $F$-regular and log Fano varieties (to appear in Adv. Math. arXiv:0905.0404)

[25] Sharp, R. Y. Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure, Trans. Amer. Math. Soc., Volume 359 (2007) no. 9, p. 4237-4258 (electronic) | DOI | MR | Zbl

[26] Smith, K. E. Test ideals in local rings, Trans. Amer. Math. Soc., Volume 347 (1995) no. 9, pp. 3453-3472 | DOI | MR | Zbl

[27] Smith, K. E. Vanishing, singularities and effective bounds via prime characteristic local algebra, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997 p. 289–325. MR1492526 (99a:14026) | MR | Zbl

[28] Smith, K. E. Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J., Volume 48 (2000), pp. 553-572 Dedicated to William Fulton on the occasion of his 60th birthday. MR1786505 (2001k:13007) | DOI | MR | Zbl

[29] Swan, R. G. On seminormality, J. Algebra, Volume 67 (1980) no. 1, pp. 210-229 MR595029 (82d:13006) | DOI | MR | Zbl