Log Canonical Threshold

The log canonical threshold of a nonzero ideal on a Kawamata log terminal pair is the greatest nonnegative coefficient for which adjoining its vanishing retains log canonicity. In valuation form it is

where ranges over prime divisors over . The ideal order is the smallest order among local generators. For the divisor on a smooth curve with zero boundary, the threshold is .

A uniform threshold through a prescribed base point requires choosing a Cartier equation whose pullback orders are controlled along the entire fiber. A discrepancy lower bound alone does not bound those orders for an arbitrary chosen equation.