On Boundedness of Divisors Computing Minimal Log Discrepancies for Surfaces

Let Γ be a finite set, and X x a fixed kawamata log terminal germ. For any lc germ (X x,B:=∑ ᵢ bᵢBᵢ) , such that bᵢ∈ Γ , Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over X x , such that a(E,X,B)=mld(X x,B) , and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that X x is not necessarily fixed and Γ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E .