Minimal model program for excellent surfaces
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 345-376.

We establish the minimal model program for log canonical and Q-factorial surfaces over excellent base schemes.

Nous prouvons les résultats prédits par le programme des modèles minimaux pour des surfaces log canoniques et Q-factorielles sur des schémas excellents.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3163
Classification: 14E30
Keywords: Minimal models, excellent surfaces, log canonical
Tanaka, Hiromu 1

1 Imperial College, London, Department of Mathematics, 180 Queen’s Gate, London SW7 2AZ (UK)
License: CC-BY-ND 4.0
@article{AIF_2018__68_1_345_0,
     author = {Tanaka, Hiromu},
     title = {Minimal model program for excellent surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {345--376},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3163},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3163/}
}
TY  - JOUR
TI  - Minimal model program for excellent surfaces
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 2018///
SP  - 345
EP  - 376
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3163/
UR  - https://doi.org/10.5802/aif.3163
DO  - 10.5802/aif.3163
LA  - en
ID  - AIF_2018__68_1_345_0
ER  - 
%0 Journal Article
%T Minimal model program for excellent surfaces
%J Annales de l'Institut Fourier
%D 2018
%P 345-376
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3163
%R 10.5802/aif.3163
%G en
%F AIF_2018__68_1_345_0
Tanaka, Hiromu. Minimal model program for excellent surfaces. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 345-376. doi : 10.5802/aif.3163. https://aif.centre-mersenne.org/articles/10.5802/aif.3163/

[1] Bădescu, Lucian Algebraic surfaces, Universitext, Springer, 2001, xii+258 pages (Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author) | DOI | MR | Zbl

[2] Beauville, Arnaud Complex algebraic surfaces, London Mathematical Society Lecture Note Series, Volume 68, Cambridge University Press, 1983, iv+132 pages (Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid) | MR | Zbl

[3] Birkar, Caucher; Chen, Yifei; Zhang, Lei Iitaka’s C n,m conjecture for 3-folds over finite fields (2015) (https://arxiv.org/abs/1507.08760v2)

[4] Cascini, Paolo; Tanaka, Hiromu; Xu, Chenyang On base point freeness in positive characteristic, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 5, pp. 1239-1272 | DOI | MR | Zbl

[5] Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, Volume 123, American Mathematical Society, 2005, x+339 pages | MR | Zbl

[6] Fujino, Osamu Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci., Volume 47 (2011) no. 3, pp. 727-789 | DOI | MR | Zbl

[7] Fujino, Osamu Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci., Volume 48 (2012) no. 2, pp. 339-371 | DOI | MR | Zbl

[8] Fujino, Osamu; Tanaka, Hiromu On log surfaces, Proc. Japan Acad. Ser. A, Volume 88 (2012) no. 8, pp. 109-114 | DOI | MR | Zbl

[9] Hacon, Christopher D.; Xu, Chenyang On finiteness of B-representations and semi-log canonical abundance, Minimal models and extremal rays (Kyoto, 2011) (Advanced Studies in Pure Mathematics) Volume 70, Mathematical Society of Japan, 2016, pp. 361-377 | MR | Zbl

[10] Hartshorne, Robin Residues and duality, Lecture Notes in Mathematics, Volume 20, Springer, 1966, vii+423 pages (with an appendix by P. Deligne) | MR | Zbl

[11] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, Volume 52, Springer, 1977, xvi+496 pages | MR | Zbl

[12] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics) Volume 10, North-Holland, 1987, pp. 283-360 | MR | Zbl

[13] Keel, Seán Basepoint freeness for nef and big line bundles in positive characteristic, Ann. Math., Volume 149 (1999) no. 1, pp. 253-286 | DOI | MR | Zbl

[14] Keeler, Dennis S. Ample filters of invertible sheaves, J. Algebra, Volume 259 (2003) no. 1, pp. 243-283 | DOI | MR | Zbl

[15] Kleiman, Steven L. Toward a numerical theory of ampleness, Ann. Math., Volume 84 (1966), pp. 293-344 | DOI | MR | Zbl

[16] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, Volume 200, Cambridge University Press, 2013, x+370 pages (With a collaboration of Sándor Kovács) | DOI | MR | Zbl

[17] Kollár, János; Kovács, Sándor Birational geometry of log surfaces (preprint available at https://sites.math.washington.edu/~kovacs/pdf/BiratLogSurf.pdf)

[18] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134, Cambridge University Press, 1998, viii+254 pages (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | DOI | MR | Zbl

[19] Lazarsfeld, Robert Positivity in algebraic geometry. I: Classical setting: Line bundles and linear series, Springer, 2004, xviii+387 pages | Zbl

[20] Lazarsfeld, Robert Positivity in algebraic geometry. II: Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge., Volume 49, Springer, 2004, xvii+385 pages | DOI | MR | Zbl

[21] Lipman, Joseph Desingularization of two-dimensional schemes, Ann. Math., Volume 107 (1978) no. 1, pp. 151-207 | DOI | MR | Zbl

[22] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, Volume 6, Oxford University Press, 2002, xvi+576 pages (Translated from the French by Reinie Erné) | MR | Zbl

[23] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8, Cambridge University Press, 1989, xiv+320 pages (Translated from the Japanese by M. Reid) | MR | Zbl

[24] Miyanishi, Masayoshi Noncomplete algebraic surfaces, Lecture Notes in Mathematics, Volume 857, Springer, 1981, xviii+244 pages | MR | Zbl

[25] Sakai, Fumio Classification of normal surfaces, Algebraic geometry, Bowdoin 1985 (Brunswick, 1985) (Proceedings of Symposia in Pure Mathematics) Volume 46, American Mathematical Society, 1987, pp. 451-465 | MR | Zbl

[26] Schwede, Karl F-adjunction, Algebra Number Theory, Volume 3 (2009) no. 8, pp. 907-950 | DOI | MR | Zbl

[27] Schwede, Karl; Tucker, Kevin On the behavior of test ideals under finite morphisms, J. Algebr. Geom., Volume 23 (2014) no. 3, pp. 399-443 | DOI | MR | Zbl

[28] Seidenberg, Abraham The hyperplane sections of normal varieties, Trans. Am. Math. Soc., Volume 69 (1950), pp. 357-386 | DOI | MR | Zbl

[29] Shafarevich, I. R. Lectures on minimal models and birational transformations of two dimensional schemes, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics, Volume 37, Tata Institute of Fundamental Research, 1966, iv+175 pages | MR | Zbl

[30] Tanaka, Hiromu Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J., Volume 216 (2014), pp. 1-70 | DOI | MR | Zbl

[31] Tanaka, Hiromu The X-method for klt surfaces in positive characteristic, J. Algebr. Geom., Volume 24 (2015) no. 4, pp. 605-628 | DOI | MR | Zbl

[32] Tanaka, Hiromu Behavior of canonical divisors under purely inseparable base changes (2016) (https://arxiv.org/abs/1502.01381v4, to appear in J. Reine Angew. Math.)

[33] Tanaka, Hiromu Pathologies on Mori fibre spaces in positive characteristic (2016) (https://arxiv.org/abs/1609.00574v2)

Cited by Sources: