Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points

Graf, Patrick; Schwede, Karl

doi:10.1090/proc/13739

Discreteness of $F$-jumping numbers at isolated non-$\mathbb {Q}$-Gorenstein points
HTML articles powered by AMS MathViewer

by Patrick Graf and Karl Schwede PDF

Proc. Amer. Math. Soc. 146 (2018), 473-487 Request permission

Abstract:

We show that the $F$-jumping numbers of a pair $(X, \mathfrak {a})$ in positive characteristic have no limit points whenever the symbolic Rees algebra of $-K_X$ is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne–Speiser–Lyubeznik–Gabber stabilization theorem describing the non-$F$-pure locus of a variety.

References

Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
Manuel Blickle, Test ideals via algebras of $p^{-e}$-linear maps, J. Algebraic Geom. 22 (2013), no. 1, 49–83. MR 2993047, DOI 10.1090/S1056-3911-2012-00576-1
Manuel Blickle and Gebhard Böckle, Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85–123. MR 2863904, DOI 10.1515/CRELLE.2011.087
Manuel Blickle, Mircea Mustaţǎ, and Karen E. Smith, Discreteness and rationality of $F$-thresholds, Michigan Math. J. 57 (2008), 43–61. Special volume in honor of Melvin Hochster. MR 2492440, DOI 10.1307/mmj/1220879396
Manuel Blickle, Mircea Mustaţă, and Karen E. Smith, $F$-thresholds of hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6549–6565. MR 2538604, DOI 10.1090/S0002-9947-09-04719-9
Manuel Blickle, Karl Schwede, Shunsuke Takagi, and Wenliang Zhang, Discreteness and rationality of $F$-jumping numbers on singular varieties, Math. Ann. 347 (2010), no. 4, 917–949. MR 2658149, DOI 10.1007/s00208-009-0461-2
S. Boucksom, T. de Fernex, C. Favre, and S. Urbinati, Valuation spaces and multiplier ideals on singular varieties, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 29–51. MR 3380442
A. Chiecchio, F. Enescu, L. E. Miller, and K. Schwede, Test ideals in rings with finitely generated anti-canonical algebras, ArXiv e-prints (2014).
Tommaso de Fernex and Christopher D. Hacon, Singularities on normal varieties, Compos. Math. 145 (2009), no. 2, 393–414. MR 2501423, DOI 10.1112/S0010437X09003996
Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin, Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469–506. MR 2068967, DOI 10.1215/S0012-7094-04-12333-4
Takao Fujita, Vanishing theorems for semipositive line bundles, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 519–528. MR 726440, DOI 10.1007/BFb0099977
Ofer Gabber, Notes on some $t$-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 711–734. MR 2099084
S. Goto, M. Herrmann, K. Nishida, and O. Villamayor, On the structure of Noetherian symbolic Rees algebras, Manuscripta Math. 67 (1990), no. 2, 197–225. MR 1042238, DOI 10.1007/BF02568430
Patrick Graf, The jumping coefficients of non-$\Bbb {Q}$-Gorenstein multiplier ideals, J. Algebra 450 (2016), 323–348. MR 3449696, DOI 10.1016/j.jalgebra.2015.11.024
Nobuo Hara, F-pure thresholds and F-jumping exponents in dimension two, Math. Res. Lett. 13 (2006), no. 5-6, 747–760. With an appendix by Paul Monsky. MR 2280772, DOI 10.4310/MRL.2006.v13.n5.a6
Nobuo Hara and Shunsuke Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 59–74. MR 2085311, DOI 10.1017/S0027763000008904
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic $p$, Ann. of Math. (2) 105 (1977), no. 1, 45–79. MR 441962, DOI 10.2307/1971025
Mordechai Katzman, Gennady Lyubeznik, and Wenliang Zhang, On the discreteness and rationality of $F$-jumping coefficients, J. Algebra 322 (2009), no. 9, 3238–3247. MR 2567418, DOI 10.1016/j.jalgebra.2008.11.032
Mordechai Katzman, Karl Schwede, Anurag K. Singh, and Wenliang Zhang, Rings of Frobenius operators, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 1, 151–167. MR 3211813, DOI 10.1017/S0305004114000176
Mordechai Katzman and Wenliang Zhang, Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3519–3533. MR 3192605, DOI 10.1090/S0002-9947-2014-05918-7
Dennis S. Keeler, Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), no. 5, 1327–1336. MR 2450210, DOI 10.1353/ajm.0.0015
János Kollár, Exercises in the birational geometry of algebraic varieties, Analytic and algebraic geometry, IAS/Park City Math. Ser., vol. 17, Amer. Math. Soc., Providence, RI, 2010, pp. 495–524. MR 2743822, DOI 10.1090/pcms/017/09
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
Robert Lazarsfeld, 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], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
Robert Lazarsfeld, 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], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI 10.2307/1971141
Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
Gennady Lyubeznik, $F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130. MR 1476089, DOI 10.1515/crll.1997.491.65
Mircea Mustaţă, The non-nef locus in positive characteristic, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 535–551. MR 3114955
Karl Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87. MR 3276163, DOI 10.1515/crelle-2012-0087
Karl Schwede, A note on discreteness of $F$-jumping numbers, Proc. Amer. Math. Soc. 139 (2011), no. 11, 3895–3901. MR 2823035, DOI 10.1090/S0002-9939-2011-10825-6
Karl Schwede, Test ideals in non-$\Bbb {Q}$-Gorenstein rings, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5925–5941. MR 2817415, DOI 10.1090/S0002-9947-2011-05297-9
Karl Schwede and Kevin Tucker, Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems, J. Math. Pures Appl. (9) 102 (2014), no. 5, 891–929. MR 3271293, DOI 10.1016/j.matpur.2014.02.009
Stefano Urbinati, Discrepancies of non-$\Bbb Q$-Gorenstein varieties, Michigan Math. J. 61 (2012), no. 2, 265–277. MR 2944480, DOI 10.1307/mmj/1339011527