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

Abstract

Fix a subset $I\subseteq \mathbb R_{>0}$ such that $γ=\inf\{ \sum_{i}n_ib_i-1>0 \mid n_i\in \mathbb Z_{\geq 0}, b_i\in I \}>0$. We give a explicit upper bound $\ell(γ)\in O(1/γ^2)$ as $γ\to 0$, 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 $k_E+1\leq \ell(γ)$. Some examples indicate that our bound is optimal.