On equivalent conjectures for minimal log discrepancies on smooth threefolds

@article{Kawakita2020OnEC,
  title={On equivalent conjectures for minimal log discrepancies on smooth threefolds},
  author={Masayuki Kawakita},
  journal={Journal of Algebraic Geometry},
  year={2020}
}
  • M. Kawakita
  • Published 7 March 2018
  • Mathematics
  • Journal of Algebraic Geometry
On smooth varieties, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. In dimension three, we reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one. 
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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
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