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Nov 9, 2018 at 3:24 comment added Takumi Murayama @be928 Glad to help! I think those extra references still have the bound $-K_X \cdot C \le 2\dim X$, though. For example, the main theorem in Cone theorems and cyclic covers has $-K_X \cdot C \le 2\dim X$.
Nov 9, 2018 at 3:14 history edited Takumi Murayama CC BY-SA 4.0
Fixing citations
Nov 9, 2018 at 2:24 history edited Takumi Murayama CC BY-SA 4.0
Fixing a small error
Nov 9, 2018 at 0:14 comment added be928 Thank you! I checked the reference and followed the remark to find that the bound $-K_X \cdot C \le \dim X + 1$ is even guaranteed when $X$ is a finite quotient of a hypersurface singularity, so in particular all terminal threefolds have this bound.
Nov 9, 2018 at 0:11 vote accept be928
Nov 8, 2018 at 20:58 history answered Takumi Murayama CC BY-SA 4.0