Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

Abstract

Fix a subset $I\subseteq {\mathbb{R}}_{>0}$ such that $\gamma =\inf \{{\sum }_i{n}_i{b}_i-1>0\mid n_i\in {\mathbb{Z}}_{\ge 0},b_i\in I\}>0$. We give a explicit upper bound $\ell (\gamma )\in O(1/{\gamma }^2)$ as $\gamma \to 0$, such that for any smooth surface $A$ of arbitrary characteristic with a closed point 0 and an $\mathbb{R}$-ideal $\mathfrak{a}$ with exponents in $I$, there always exists a prime divisor $E$ over $A$ computing the minimal log discrepancy of $(A,\mathfrak{a})$ at 0 and with its log discrepancy $k_E+1\le \ell (\gamma )$. Some examples indicate that our bound is optimal.

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