Journal of the
Korean Mathematical Society JKMS

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 K"ahler--Einstein metric, then it should admit a special cascade and it satisfies the equality of the orbifold Bogomolov--Miyaoka--Yau inequality, i.e., $K^2 = 3e_{orb}.$ 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: Toric log del Pezzo surface, $mathbb{P}^1$-fibration, cascade

MSC numbers: Primary 14M25; Secondary 14J45, 14J26, 52B20

Supported by: 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).