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

TL;DR

This paper extends Kollár's method for computing log canonical thresholds using weighted blowups to broader cases, relating thresholds to Newton polyhedra, and generalizes Varchenko's theorem to non-isolated singularities.

Contribution

It generalizes Kollár's proposition to non-negative weights and extends Varchenko's theorem to non-isolated singularities in plane curves.

Findings

Log canonical threshold is at most 1/c for points on Newton polyhedron facets.

Equality holds under certain normalizations or facet conditions in two dimensions.

Generalizes previous results to non-isolated singularities.

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.