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

TL;DR

This paper establishes an explicit upper bound on the discrepancies of divisors computing minimal log discrepancies on surfaces, which is significant for understanding singularities in algebraic geometry.

Contribution

The paper provides a new explicit upper bound for discrepancies on surfaces, improving understanding of minimal log discrepancies and their computation.

Findings

Bound is proportional to 1/γ^2 as γ approaches 0

Existence of a prime divisor with discrepancy bounded by ℓ(γ)

Examples suggest the bound is optimal

Abstract

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