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

Title & Authors
ON THE ORBIFOLD EULER CHARACTERISTIC OF LOG DEL PEZZO SURFACES OF RANK ONE
Hwang, DongSeon;

Abstract
It is known that the orbifold Euler characteristic $\small{e_{orb}(S)}$ of a log del Pezzo surface S of rank one satisfies the inequality $\small{0{\leq}e_{orb}(S){\leq}3}$. In this note, we show that the orbifold Euler characteristic of S is strictly positive, i.e., 0 < $\small{e_{orb}(S)}$. 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;
Language
English
Cited by
References
1.
G. N. Belousov, Del Pezzo surfaces with log terminal singularities, Math. Notes 83 (2008), no. 1-2, 152-161.

2.
E. Brieskorn, Rationale Singularitaten komplexer Flachen, Invent. Math. 4 (1968), 336-358.

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.

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.

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.

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.

10.
H. Kojima and T. Takahashi, Normal del Pezzo surfaces of rank one with log canonical singularities, J. Algebra 360 (2012), 53-70.

11.
Q. Ye, On Gorenstein log del Pezzo surfaces, Japan. J. Math. (N.S.) 28 (2002), no. 1, 87-136.

12.
D. Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), no. 2, 461-497.