Accumulation points on 3-fold canonical thresholds

Abstract

Let 𝑘 ≥ 2 be a given integer. We study the set of 3-fold canonical thresholds ct(𝑋;𝑆) with \frac{1}{𝑘} < ct(𝑋;𝑆) < \frac{1}{𝑘−1} where 𝑆 is a ℚ-Cartier prime divisor of a projective 3-fold 𝑋. Express ct(𝑋;𝑆) as the rational number \frac{𝑎}{𝑚} where 𝑎 (resp. 𝑚) denotes the weighted discrepancy (resp. weighted multiplicity). We conclude that if 𝑎 ≥ 54𝑘⁴, then we may choose positive integers 𝑝 and 𝑞 satisfying ct(𝑋;𝑆) = \frac{𝑎}{𝑚} = \frac{1}{𝑘} + \frac{𝑞}{𝑝} and 𝑞 < 6𝑘³. As a consequence, the set of accumulation points of the set of 3-fold canonical thresholds consists of {0} ∪ { \frac{1}{𝑘} }_{𝑘∈ℤ _{≥ 2}}. Moreover, we generalize the ACC for the set of 3-fold canonical thresholds to pairs.