The degrees of plane curves with prescribed log canonical threshold
type: Item
instance of: scholarly article
title: The degrees of plane curves with prescribed log canonical threshold
zbMATH Open document ID: 1195.14045
author: Kyungyong Lee
published in: Journal of Algebra
publication date: 2010-03-04T00:00:00Z
review text: The author obtains a bound on the degree of curves with fixed log canonical threshold at very general points on \(\mathbb{P}^2\). There has been a considerable amount of research on the existence of plane curves of a given degree with prescribed singularities from different points of view. In particular, the multiplicity and the Milnor number have been mainly used as a measures of singularity. Another invariant, studied recently in birational geometry, can be considered: the log canonical threshold. Let \(X\) be a smooth complex surface, \(p\) a point on \(X\), and \(C \subset X\) a (possibly reducible and non-reduced) curve passing through \(p\). Let \(S\) be a local analytical neighborhood of \(X\) at \(p\). If \(C=\sum_i a_i C_i\), with \(C_i\) prime divisors, then the component multiplicity of \(C\) at \(p\) is defined as comp-mult\(_p(C )= \max\{a_i | p \in C_i\}\). Let \(e\) be the component multiplicity of \(C\) at \(p\). Consider the minimal log resolution \(\mu:S' \to S\) of \((S,C)\), which consists of smooth blow-ups. Such a resolution exists and is uniquely obtained when we blow up as few time as possible. Let \(K_{S'/S}=\sum_i a_i F_i \) and \(\mu^*C=\sum_i b_i F_i + (\text{proper transform of }C)\), where \(i\)'s denote the order of smooth blow-ups and \(F_i\) are the corresponding irreducible exceptional divisors. Then the log canonical threshold of \(C\) at \(p\) is \[ \text{lct}(C;p)=\text{lct}(S,C;p)=\min\{\min_i \frac{a_i+1}{b_i}, \frac{1}{e}\} \] The locus of log canonical singularities, or the \(LC-locus\) for short, of \(C\) at \(p\) is the zero locus of \(\mathcal{J}(\text{lct}( C ) \cdot C)\). If \(\text{lct}(C )=\frac{1}{e}\) then the LC-locus is \(1\)-dimensional, while if \(\text{lct}(C )=\min_i \frac{a_i+1}{b_i} <\frac{1}{e}\) then the LC-locus is \(0\)-dimensional. If \(C\) has the same multiplicity, the same log canonical threshold and the same component multiplicity at each of the several points \(p_1, \dots, p_r\) then we say that \(C…
reviewed by: Cristiano Bocci
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 5677310
zbMATH Keywords: plane curves
zbMATH Keywords: log canonical threshold
zbMATH Keywords: multiplier ideals
zbMATH Keywords: colength
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.09.016
OpenAlex ID: W2018080892
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DOI: 10.1016/J.JALGEBRA.2009.09.016
description: scientific article; zbMATH DE number 5677310