• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.179-190
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• 2022

A cascade of toric log del Pezzo surfaces of Picard number one was introduced as a language of classifying all such surfaces. In this paper, we introduce a generalized concept, a semicascade of toric log del Pezzo surfaces. As applications, we discuss Kähler-Einstein toric log del Pezzo surfaces and derive a bound on the Picard number in terms of the number of singular points, generalizing some results of Dais and Suyama.

• Journal of the Korean Mathematical Society
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• v.61 no.5
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• pp.837-851
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• 2024

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 = 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.