K-stable del Pezzo surfaces with a single quotient singularity of type 1/(mn-1)(1,n) and two exceptional curves in the minimal resolution are exactly S^4_{2,2}, S^5_{2,2}, S^5_{3,2}, S^6_{3,2}, S^6_{4,2}, S^7_{4,2}, S^5_{3,3}, S^6_{4,3}; S^3_{2,2} and S^7_{5,2} are strictly semistable.
Anticanonical divisor with good asymptotic base loci
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In this paper, we give a characterization of Fano type varieties in terms of the asymptotic base loci of $-(K_X+\Delta)$. We also show that for a potentially lc pair $(X,\Delta)$, if no plc centers are contained in the augmented base locus $\mathbf{B}_{+}(-(K_X+\Delta))$, then $(X,\Delta)$ has a good $-(K_X+\Delta)$-minimal model. This gives an analogous result of Birkar--Hu on the existence of good minimal models.
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K-stability of del Pezzo surfaces with a single quotient singularity
K-stable del Pezzo surfaces with a single quotient singularity of type 1/(mn-1)(1,n) and two exceptional curves in the minimal resolution are exactly S^4_{2,2}, S^5_{2,2}, S^5_{3,2}, S^6_{3,2}, S^6_{4,2}, S^7_{4,2}, S^5_{3,3}, S^6_{4,3}; S^3_{2,2} and S^7_{5,2} are strictly semistable.