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This paper shows that Mustata-Nakamura's conjecture holds for pairsconsisting of a smooth surface and a multiideal with a real exponent over thebase field of positive characteristic. As corollaries, we obtain the ascendingchain condition of the minimal log discrepancies and of the log canonicalthresholds for those pairs. We also obtain finiteness of the set of the minimallog discrepancies of those pairs for a fixed real exponent.

The minimal log discrepancies on a smooth surface in positive characteristic