DOI QR코드

DOI QR Code

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

  • 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

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. 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
  2. E. Brieskorn, Rationale Singularitaten komplexer Flachen, Invent. Math. 4 (1968), 336-358. https://doi.org/10.1007/BF01425318
  3. 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.
  4. 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
  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. 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
  11. 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
  12. D. Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), no. 2, 461-497.