On log canonical rational singularities. (English) Zbl 1364.14013

Fujino, Osamu

Proc. Japan Acad., Ser. A 92, No. 1, 13-18 (2016).

In the paper under review, the author proves several results concerning the minimal model program for varieties with bad singularities, i.e. log canonical singularities, following the analysis in [O. Fujino, Publ. Res. Inst. Math. Sci. 47, No. 3, 727–789 (2011; Zbl 1234.14013)]. In particular, in the main theorems the authour proves that the class of rational log canonical singularities is stable under the main operations of the minimal models program, i.e. divisorial and Fano contractions in Theorem 1.1 and flips in Theorem 1.2. We should note that the singularities of \(X\) are not automatically rational when \((X, \Delta)\) is a log canonical pair. Moreover, even if we start from a log canonical pair, its log canonical model might not have rational singularities, see Example 5.1. In Theorem 4.1, the author gives a supplementary result in the case of log canonical surfaces. In particular he shows that given a log canonical surface \((X, \Delta)\) and \(f : X \to Y\) a projective birational morphism such that \(Y\) is normal and \(-(K_X+\Delta)\) is \(f\)-ample, then the exceptional locus of \(f\) passes through no nonrational singular points of \(X\). In particular, the number of nonrational log canonical singularities never decreases under a minimal model program, see Corollary 4.2.

Reviewer: Diletta Martinelli (Edinburgh)

MSC:

14E30	Minimal model program (Mori theory, extremal rays)
14J17	Singularities of surfaces or higher-dimensional varieties

Keywords:

rational singularities; log canonical singularities; minimal model program; log canonical surfaces

Citations:

Zbl 1234.14013

References:

[1]	V. Alexeev and C. D. Hacon, Non-rational centers of log canonical singularities, J. Algebra 369 (2012), 1-15. · Zbl 1275.14004 · doi:10.1016/j.jalgebra.2012.06.015
[2]	C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 325-368. · Zbl 1256.14012 · doi:10.1007/s10240-012-0039-5
[3]	O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727-789. · Zbl 1234.14013 · doi:10.2977/PRIMS/50
[4]	O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339-371. · Zbl 1248.14018 · doi:10.2977/PRIMS/71
[5]	O. Fujino, Foundation of the minimal model program, 2014/4/16, version 0.01. (Preprint).
[6]	O. Fujino, Some remarks on the minimal model program for log canonical pairs, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, 149-192. · Zbl 1435.14017
[7]	O. Fujino, On semipositivity, injectivity, and vanishing theorems. (Preprint). · Zbl 1189.14024 · doi:10.3792/pjaa.85.95
[8]	O. Fujino and H. Tanaka, On log surfaces, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 8, 109-114. · Zbl 1268.14012 · doi:10.3792/pjaa.88.109
[9]	C. D. Hacon and C. Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161-195. · Zbl 1282.14027 · doi:10.1007/s00222-012-0409-0
[10]	J. Kollár and S. Mori, Birational geometry of algebraic varieties , translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, Cambridge, 1998.
[11]	S. J. Kovács, A characterization of rational singularities, Duke Math. J. 102 (2000), no. 2, 187-191. · Zbl 0973.14001 · doi:10.1215/S0012-7094-00-10221-9
[12]	J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195-279. · Zbl 0181.48903 · doi:10.1007/BF02684604
[13]	H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1-70. · Zbl 1311.14020 · doi:10.1215/00277630-2801646