Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron

Abstract

There is a proposition due to Koll\'ar 1997 on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most $1/c$, where $(c, \ldots, c)$ is a point on a facet of its Newton polyhedron. Moreover, in the case $n = 2$, if the power series is weakly normalised with respect to this facet or the point $(c, c)$ belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities.