Lowest log canonical thresholds of a reduced plane curve of degree d

Abstract

We describe the sixth worst singularity that a plane curve of degree dgeqslant 5 could have, using its log canonical threshold at the point of singularity. This is an extension of a result due to Cheltsov (J Geom Anal 27(3):2302–2338, 2017) wherein the five lowest values of log canonical thresholds of a plane curve of degree d geqslant 3 were computed. These six small log canonical thresholds, in order, are 2 / d, ({2d-3})/{(d-1)^2}, ({2d-1})/(d^2-d), ({2d-5})/({d^2-3d+1}), ({2d-3})/(d^2-2d) and ({2d-7})/({d^2-4d+1}). We give examples of curves with these values as their log canonical thresholds using illustrations.