Number of Singular Points on Projective Surfaces

Abstract

The number of singular points on a klt Fano surface X is less than or equal to 2ρ(X) + 2.

Keywords

Singular points / Fano variety / Canonical bundle formula / 14E30 / 14B05

References

[1] BelousovG N. Del Pezzo surfaces with log terminal singularities. Mat. Zametki, 2008, 83(2): 170-180

[2] BelousovG N. The maximal number of singular points on log del Pezzo surfaces. J. Math. Sci. Univ. Tokyo, 2009, 16(2): 231-238

[3] BirkarC. On connectedness of non-klt loci of singularities of pairs. J. Differential Geom., 2024, 126(2): 431-474.

[4] BirkarC, CasciniP, HaconC D, McKernanJ. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 2010, 23(2): 405-468.

[5] BirkarC, ZhangD Q. Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs. Pub. Math. IHES., 2016, 123: 283-331.

[6] BlacheR. The structure of l.c. surfaces of Kodaira dimension zero, I. J. Algebraic Geom., 1995, 4(1): 137-179

[7] BrownM, McKernanJ, SvaldiR, ZongH R. A geometric characterization of toric varieties. Duke Math. J., 2018, 167(5): 923-968.

[8] ChenM, JiangC, LiB. On minimal varieties growing from quasismooth weighted hypersurfaces. J. Differential Geom., 2024, 127(1): 35-76.

[9] DolgachevIWeighted projective varieties, Group Actions and Vector Fields (Vancouver, B.C.), 1982, Berlin. Springer-Verlag. 956

[10] FilipazziS. On a generalized canonical bundle formula and generalized adjunction. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2020, 5(21): 1187-1221

[11] Filipazzi, S., and Svaldi, R., On the connectedness principle and dual complexes for generalized pairs, Forum Math. Sigma, 11, 2023, Paper No. e33, 39pp.

[12] FurushimaM. Singular del Pezzo surfaces and analytic compactifiations of 3-dimensional complex affine space ℂ³. Nagoya Math. J., 1986, 104: 1-28.

[13] HaconC D, HanJ. On a connectedness principle of Shokurov-Kollár type. Sci. China Math., 2019, 62(3): 411-416.

[14] Han, J., Liu, J. and Shokurov, V. V., ACC for minimal log discrepancies for exceptional singularities, https://doi.org/10.1007/s42543-024-00091-x.

[15] HanJ, LiuW. On numerical nonvanishing for generalized log canonical pairs. Doc. Math., 2020, 25: 93-123.

[16] HanJ, LiuW. On a generalized canonical bundle formula for generically finite morphisms. Annales de l’Institut Fourier, 2021, 71(5): 2047-2077.

[17] HanJ, LuoY. On boundedness of divisors computing minimal log discrepancies for surfaces. J. Inst. Math. Jussieu, 2023, 22(6): 2907-2930.

[18] HartshorneRAlgebraic geometry, 1997, New York-Heidelberg. Springer-Verlag. 52

[19] Iano-FletcherA RWorking with weighted complete intersections, Explicit Birational Geometry of 3-folds, 2000, Cambridge. Cambridge Univ. Press. 281

[20] Keel, S., and McKernan, J., Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc., 140(669), 1999.

[21] KollárJ, MoriSBirational geometry of algebraic varieties, 1998, Cambridge. Cambridge Univ. Press. . 134

[22] MatsumuraHCommutative ring theory, 1989, Cambridge. Cambridge Univ. Press. 8

[23] MiyanishiM, ZhangD-Q. Gorenstein log del Pezzo surfaces of rank one. J. Algebra, 1988, 118(1): 63-64.

[24] PetersC. On the maximal number of du Val singularities for a K3 surface. Geom. Dedicata, 2021, 214: 383-388.

[25] ProkhorovYLectures on complements on log surfaces, 2001, Tokyo. Mathematical Society of Japan. 10

[26] ShokurovV V. 3-fold log flips. Izv. Ross. Akad. Nauk Ser. Mat., 1992, 56: 105-203

[27] ZhangD-Q. On Iitaka surfaces. Osaka J. Math., 1987, 46: 435-450

[28] ZhangD-Q. Logarithmic del Pezzo surfaces of rank one with contractible boundaries. Osaka J. Math., 1988, 25(2): 461-497

[29] ZhangD-Q. Logarithmic del Pezzo surfacees one with rational double and triple singular points. Tohoku Math. J., 1989, 41(3): 399-452. 2)

[30] ZhangD-Q. Logarithmic Enriques surfaces. J. Math. Kyoto Univ., 1991, 31(2): 419-466

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