Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CASCADES OF TORIC LOG DEL PEZZO SURFACES OF PICARD NUMBER ONE

  • DongSeon Hwang (Center for Complex Geometry Institute for Basic Science (IBS))
  • Received : 2021.02.04
  • Accepted : 2024.04.26
  • Published : 2024.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

We classify toric log del Pezzo surfaces of Picard number one by introducing the notion, cascades. As an application, we show that if such a surface admits a Kähler-Einstein metric, then it should admit a special cascade and it satisfies the equality of the orbifold Bogomolov-Miyaoka-Yau inequality, i.e., K2 = 3eorb. Moreover, we provide an algorithm to compute a toric log del Pezzo surfaces of Picard number one for a given input of singularity types.

Keywords

Acknowledgement

The author would like to express his deep gratitude to the referee for the careful reading and suggestions to improve the exposition of this paper. The author was supported by Samsung Science and Technology Foundation under Project SSTF-BA1602-03, the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2021R1A2C1093787), and the Institute for Basic Science (IBS-R032-D1).

References

  1. V. Alekseev and V. V. Nikulin, Del Pezzo and K3 surfaces, MSJ Memoirs, 15, Math. Soc. Japan, Tokyo, 2006. https://doi.org/10.1142/e002
  2. R. J. Berman and B. Berndtsson, Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 649-711. https://doi.org/10.5802/afst.1386
  3. K. Chan and N. C. Leung, Miyaoka-Yau-type inequalities for Kahler-Einstein manifolds, Comm. Anal. Geom. 15 (2007), no. 2, 359-379.
  4. J.-L. Colliot-Thelene and A. N. Skorobogatov, The Brauer-Grothendieck Group, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 71, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-74248-5
  5. D. I. Dais, Geometric combinatorics in the study of compact toric surfaces, Algebraic and geometric combinatorics, 71-123, Contemp. Math. 423, Amer. Math. Soc., Providence, RI, 2006. https://doi.org/10.1090/conm/423/08076
  6. D. I. Dais, Classification of toric log del Pezzo surfaces having Picard number 1 and index ≤ 3, Results Math. 54 (2009), no. 3-4, 219-252. https://doi.org/10.1007/s00025-009-0375-z
  7. D. I. Dais, Toric log del Pezzo surfaces with one singularity, Adv. Geom. 20 (2020), no. 1, 121-138. https://doi.org/10.1515/advgeom-2019-0009
  8. D. I. Dais and B. Nill, A boundedness result for toric log del Pezzo surfaces, Arch. Math. (Basel) 91 (2008), no. 6, 526-535. https://doi.org/10.1007/s00013-008-2913-4
  9. F. R. DeMeyer and T. J. Ford, On the Brauer group of toric varieties, Trans. Amer. Math. Soc. 335 (1993), no. 2, 559-577. https://doi.org/10.2307/2154394
  10. T. J. Ford, Every finite abelian group is the Brauer group of a ring, Proc. Amer. Math. Soc. 82 (1981), no. 3, 315-321. https://doi.org/10.2307/2043931
  11. K. Fujita and K. Yasutake, Classification of log del Pezzo surfaces of index three, J. Math. Soc. Japan 69 (2017), no. 1, 163-225. https://doi.org/10.2969/jmsj/06910163
  12. Graded Ring Database, http://www.grdb.co.uk/
  13. R. V. Gurjar, K. Masuda, and M. Miyanishi, Affine space fibrations, Polynomial rings and affine algebraic geometry, 151-193, Springer Proc. Math. Stat., 319, Springer, Cham, 2020.
  14. F. Hidaka and K. Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4 (1981), no. 2, 319-330. https://doi.org/10.3836/tjm/1270215157
  15. D. Hwang, Algebraic Montgomery-Yang problem and cascade structure, arXiv:2012.13355, preprint, 2020.
  16. D. Hwang, Semicascades of toric log del Pezzo surfaces, Bull. Korean Math. Soc. 59 (2022), no. 1, 179-190. https://doi.org/10.4134/BKMS.b210211
  17. D. Hwang and J. Keum, The maximum number of singular points on rational homology projective planes, J. Algebraic Geom. 20 (2011), no. 3, 495-523. https://doi.org/10.1090/S1056-3911-10-00532-1
  18. D. Hwang and Y. Kim, Symmetric and Kahler-Einstein Fano polygons, arXiv:2012.13373v3, preprint, 2020.
  19. D. Hwang and Y. Yoon, On Kahler-Einstein fake weighted projective spaces, Eur. J. Math. 8 (2022), no. 3, 985-990. https://doi.org/10.1007/s40879-021-00468-7
  20. G. Kapustka and M. Kapustka, Equations of log del Pezzo surfaces of index ≤ 2, Math. Z. 261 (2009), no. 1, 169-188. https://doi.org/10.1007/s00209-008-0320-y
  21. A. M. Kasprzyk, M. Kreuzer, and B. Nill, On the combinatorial classification of toric log del Pezzo surfaces, LMS J. Comput. Math. 13 (2010), 33-46. https://doi.org/10.1112/S1461157008000387
  22. A. M. Kasprzyk and B. Nill, Fano polytopes, Strings, gauge fields, and the geometry behind, 349-364, World Sci. Publ., Hackensack, NJ, 2013.
  23. S. Keel and J. McKernan, Rational Curves on Quasi-Projective Surfaces, Mem. Amer. Math. Soc. 140, no. 669, viii+153 pp, 1999. https://doi.org/10.1090/memo/0669
  24. R. Kobayashi, S. Nakamura, and F. Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 238-241. http://projecteuclid.org/euclid.pja/1195512773
  25. A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. London Math. Soc. (3) 86 (2003), no. 2, 358-396. https://doi.org/10.1112/S0024611502013874
  26. G. Megyesi, Generalisation of the Bogomolov-Miyaoka-Yau inequality to singular surfaces, Proc. London Math. Soc. (3) 78 (1999), no. 2, 241-282. https://doi.org/10.1112/S0024611599001719
  27. M. Miyanishi, Open Algebraic Surfaces, CRM Monograph Series, 12, Amer. Math. Soc., Providence, RI, 2001. https://doi.org/10.1090/crmm/012
  28. N. Nakayama, Classification of log del Pezzo surfaces of index two, J. Math. Sci. Univ. Tokyo 14 (2007), no. 3, 293-498.
  29. O. Riemenschneider, Deformationen von Quotientensingularitaten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211-248. https://doi.org/10.1007/BF01351850
  30. Y. Shi and X. Zhu, Kahler-Ricci solitons on toric Fano orbifolds, Math. Z. 271 (2012), no. 3-4, 1241-1251. https://doi.org/10.1007/s00209-011-0913-8
  31. Y. Suyama, Classification of toric log del Pezzo surfaces with few singular points, Electron. J. Combin. 28 (2021), no. 1, Paper No. 1.39, 17 pp. https://doi.org/10.37236/9056
  32. D.-Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), no. 2, 461-497. http://projecteuclid.org/euclid.ojm/1200780782