no. 6
Nakamaye’s theorem on log canonical pairs
[Le théorème de Nakamaye dans les paires log-canoniques]
Cacciola, Salvatore1 ; Lopez, Angelo Felice1
1 Dipartimento di Matematica e Fisica Università di Roma Tre Largo San Leonardo Murialdo 1 00146, Roma (Italy)
Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2283-2298.

On propose une généralisation de la description de Nakamaye, par le biais de la théorie d’intersection, du lieu de base augmenté d’un diviseur grand et nef sur une paire normale avec singularités log-canoniques ou, plus généralement, sur une variété avec lieu non-lc de dimension ≤1. On propose aussi une généralisation de la description de Ein-Lazarsfeld-Mustaţă-Nakamaye-Popa, en termes de valuations, des sous-variétés du lieu de base restreint d’un diviseur grand sur une paire normale avec singularités klt.

We generalize Nakamaye’s description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension ≤1. We also generalize Ein-Lazarsfeld-Mustaţă-Nakamaye-Popa’s description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.

DOI : 10.5802/aif.2913
Classification : 14C20, 14F18, 14E15, 14B05
Keywords: Base loci, log-canonical singularities, non-lc ideal
Mot clés : lieux de base, singularités log-canoniques, idéaux non-lc

Cacciola, Salvatore 1 ; Lopez, Angelo Felice 1

1 Dipartimento di Matematica e Fisica Università di Roma Tre Largo San Leonardo Murialdo 1 00146, Roma (Italy) 
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Cacciola, Salvatore; Lopez, Angelo Felice. Nakamaye’s theorem on log canonical pairs. Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2283-2298. doi : 10.5802/aif.2913. https://aif.centre-mersenne.org/articles/10.5802/aif.2913/

[1] Ambro, F. Quasi-log varieties, Tr. Mat. Inst. Steklova, Volume 240 (2003), pp. 220-239 (Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry) | MR | Zbl

[2] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl

[3] Boucksom, Sébastien; Broustet, Amaël; Pacienza, Gianluca Uniruledness of stable base loci of adjoint linear systems via Mori theory, Math. Z., Volume 275 (2013) no. 1-2, pp. 499-507 | DOI | MR | Zbl

[4] Boucksom, Sébastien; Favre, C.; Jonnson, M. A refinement of Izumi’s theorem (arXiv:math.AG. 1209.4104)

[5] Cacciola, S.; di Biagio, L. Asymptotic base loci on singular varieties (To appear in Math. Z. DOI 10.1007/s00209-012-1128-3; arXiv:math.AG.1105.1253) | MR | Zbl

[6] Cascini, Paolo; McKernan, James; Mustaţă, Mircea The augmented base locus in positive characteristic, Proc. Edinb. Math. Soc. (2), Volume 57 (2014) no. 1, pp. 79-87 | DOI | MR | Zbl

[7] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 6, pp. 1701-1734 | DOI | Numdam | MR | Zbl

[8] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Restricted volumes and base loci of linear series, Amer. J. Math., Volume 131 (2009) no. 3, pp. 607-651 | DOI | MR | Zbl

[9] de Fernex, Tommaso; Hacon, Christopher D. Singularities on normal varieties, Compos. Math., Volume 145 (2009) no. 2, pp. 393-414 | DOI | MR | Zbl

[10] Fujino, Osamu Theory of non-lc ideal sheaves: basic properties, Kyoto J. Math., Volume 50 (2010) no. 2, pp. 225-245 | DOI | MR | Zbl

[11] Fujita, T. A relative version of Kawamata-Viehweg vanishing theorem (1985) (Preprint Tokyo Univ.)

[12] Gibney, Angela; Keel, Sean; Morrison, Ian Towards the ample cone of M ¯ g,n , J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 273-294 | DOI | MR | Zbl

[13] Hacon, Christopher D.; McKernan, James Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006) no. 1, pp. 1-25 | DOI | MR | Zbl

[14] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | MR | Zbl

[15] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998, pp. viii+254 | Zbl

[16] Lazarsfeld, Robert Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer-Verlag, Berlin, 2004, pp. xviii+385 | MR | Zbl

[17] Lehmann, Brian On Eckl’s pseudo-effective reduction map, Trans. Amer. Math. Soc., Volume 366 (2014) no. 3, pp. 1525-1549 | DOI | MR

[18] Lesieutre, J. The diminished base locus is not always closed (arXiv:math.AG.1212.3738) | MR

[19] Mustaţă, Mircea The non-nef locus in positive characteristic, A celebration of algebraic geometry (Clay Math. Proc.), Volume 18, Amer. Math. Soc., Providence, RI, 2013, pp. 535-551 | MR

[20] Nakamaye, Michael Stable base loci of linear series, Math. Ann., Volume 318 (2000) no. 4, pp. 837-847 | DOI | MR | Zbl

[21] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004, pp. xiv+277 | MR | Zbl

[22] Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl

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