Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914014624219136 |
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| author | Paemurru, Erik |
| author_facet | Paemurru, Erik |
| contents | There is a proposition due to Kollár 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18238 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron Paemurru, Erik Algebraic Geometry Complex Variables 14H20 (Primary) 14B05, 32S25, 14Q05, 14H50 There is a proposition due to Kollár 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. |
| title | Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron |
| topic | Algebraic Geometry Complex Variables 14H20 (Primary) 14B05, 32S25, 14Q05, 14H50 |
| url | https://arxiv.org/abs/2404.18238 |