Nakamaye’s theorem on log canonical pairs
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, p. 2283-2298

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.

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.

DOI : https://doi.org/10.5802/aif.2913
Classification:  14C20,  14F18,  14E15,  14B05
Keywords: Base loci, log-canonical singularities, non-lc ideal
@article{AIF_2014__64_6_2283_0,
     author = {Cacciola, Salvatore and Lopez, Angelo Felice},
     title = {Nakamaye's theorem on log canonical pairs},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     pages = {2283-2298},
     doi = {10.5802/aif.2913},
     mrnumber = {3331167},
     zbl = {06387340},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_6_2283_0}
}
Cacciola, Salvatore; Lopez, Angelo Felice. Nakamaye’s theorem on log canonical pairs. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2283-2298. doi : 10.5802/aif.2913. http://www.numdam.org/item/AIF_2014__64_6_2283_0/

[1] Ambro, F. Quasi-log varieties, Tr. Mat. Inst. Steklova, Tome 240 (2003), pp. 220-239 (Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry) | MR 1993751 | Zbl 1081.14021

[2] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; Mckernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Tome 23 (2010) no. 2, pp. 405-468 | Article | MR 2601039 | Zbl 1210.14019

[3] Boucksom, Sébastien; Broustet, Amaël; Pacienza, Gianluca Uniruledness of stable base loci of adjoint linear systems via Mori theory, Math. Z., Tome 275 (2013) no. 1-2, pp. 499-507 | Article | MR 3101817 | Zbl 1278.14021

[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 3101802 | Zbl 1282.14011

[6] Cascini, Paolo; Mckernan, James; Mustaţă, Mircea The augmented base locus in positive characteristic, Proc. Edinb. Math. Soc. (2), Tome 57 (2014) no. 1, pp. 79-87 | Article | MR 3165013 | Zbl 1290.14006

[7] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 6, pp. 1701-1734 | Article | Numdam | MR 2282673 | Zbl 1127.14010

[8] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Restricted volumes and base loci of linear series, Amer. J. Math., Tome 131 (2009) no. 3, pp. 607-651 | Article | MR 2530849 | Zbl 1179.14006

[9] De Fernex, Tommaso; Hacon, Christopher D. Singularities on normal varieties, Compos. Math., Tome 145 (2009) no. 2, pp. 393-414 | Article | MR 2501423 | Zbl 1179.14003

[10] Fujino, Osamu Theory of non-lc ideal sheaves: basic properties, Kyoto J. Math., Tome 50 (2010) no. 2, pp. 225-245 | Article | MR 2666656 | Zbl 1200.14033

[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., Tome 15 (2002) no. 2, pp. 273-294 | Article | MR 1887636 | Zbl 0993.14009

[13] Hacon, Christopher D.; Mckernan, James Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Tome 166 (2006) no. 1, pp. 1-25 | Article | MR 2242631 | Zbl 1121.14011

[14] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, North-Holland, Amsterdam (Adv. Stud. Pure Math.) Tome 10 (1987), pp. 283-360 | MR 946243 | Zbl 0672.14006

[15] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 134 (1998), pp. viii+254 | Zbl 0926.14003

[16] Lazarsfeld, Robert Positivity in algebraic geometry. II, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 49 (2004), pp. xviii+385 | MR 2095472 | Zbl 1093.14500

[17] Lehmann, Brian On Eckl’s pseudo-effective reduction map, Trans. Amer. Math. Soc., Tome 366 (2014) no. 3, pp. 1525-1549 | Article | MR 3145741

[18] Lesieutre, J. The diminished base locus is not always closed (arXiv:math.AG.1212.3738) | MR 3269465

[19] Mustaţă, Mircea The non-nef locus in positive characteristic, A celebration of algebraic geometry, Amer. Math. Soc., Providence, RI (Clay Math. Proc.) Tome 18 (2013), pp. 535-551 | MR 3114955

[20] Nakamaye, Michael Stable base loci of linear series, Math. Ann., Tome 318 (2000) no. 4, pp. 837-847 | Article | MR 1802513 | Zbl 1063.14008

[21] Nakayama, Noboru Zariski-decomposition and abundance, Mathematical Society of Japan, Tokyo, MSJ Memoirs, Tome 14 (2004), pp. xiv+277 | MR 2104208 | Zbl 1061.14018

[22] Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. Math., Tome 165 (2006) no. 3, pp. 551-587 | Article | MR 2242627 | Zbl 1108.14031