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ON THE ORBIFOLD EULER CHARACTERISTIC OF LOG DEL PEZZO SURFACES OF RANK ONE

  • Hwang, DongSeon (Department of Mathematics Ajou University)
  • Received : 2014.04.07
  • Published : 2014.07.01

Abstract

It is known that the orbifold Euler characteristic $e_{orb}(S)$ of a log del Pezzo surface S of rank one satisfies the inequality $0{\leq}e_{orb}(S){\leq}3$. In this note, we show that the orbifold Euler characteristic of S is strictly positive, i.e., 0 < $e_{orb}(S)$. Moreover, we also show, by construction, the existence of log del Pezzo surfaces of rank one with arbitrarily small orbifold Euler characteristic.

Keywords

log del Pezzo surface of rank one;orbifold Euler characteristic

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. 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
  2. G. N. Belousov, Del Pezzo surfaces with log terminal singularities, Math. Notes 83 (2008), no. 1-2, 152-161. https://doi.org/10.1134/S0001434608010185
  3. E. Brieskorn, Rationale Singularitaten komplexer Flachen, Invent. Math. 4 (1968), 336-358. https://doi.org/10.1007/BF01425318
  4. R. V. Gurjar and D. Q. Zhang, ${\pi}_1$ of smooth points of a log del Pezzo surface is finite. I, J. Math. Sci. Univ. Tokyo 1 (1994), no. 1, 137-180.
  5. D. Hwang and J. Keum, Construction of singular rational surfaces of Picard number one with ample canonical divisor, Proc. Amer. Math. Soc. 140 (2012), no. 6, 1865-1879. https://doi.org/10.1090/S0002-9939-2011-11038-4
  6. D. Hwang and J. Keum, Algebraic Montgomery-Yang Problem: the log del Pezzo surface case, to appear in J. Math. Soc. Jpn.
  7. S. Keel and J. McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153 pp.
  8. H. Kojima, Supplement to Normal del Pezzo surfaces of rank one with log canonical singularities by H. Kojima and T. Takahashi [J. Algebra 360 (2012), 53-70], J. Algebra 377 (2013), 312-316. https://doi.org/10.1016/j.jalgebra.2012.11.028
  9. H. Kojima and T. Takahashi, Notes on minimal compactifications of the affine plane, Ann. Mat. Pura. Appl. (4) 188 (2009), no. 1, 153-169. https://doi.org/10.1007/s10231-008-0069-2
  10. Q. Ye, On Gorenstein log del Pezzo surfaces, Japan. J. Math. (N.S.) 28 (2002), no. 1, 87-136. https://doi.org/10.4099/math1924.28.87
  11. H. Kojima and T. Takahashi, Normal del Pezzo surfaces of rank one with log canonical singularities, J. Algebra 360 (2012), 53-70. https://doi.org/10.1016/j.jalgebra.2012.02.026
  12. D. Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), no. 2, 461-497.