The degrees of plane curves with prescribed log canonical threshold
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 \(C\) is almost-equisingular at \(p_1, \dots, p_r\). The main results in this paper are the following theorems. Theorem 1. Fix \(r>9\) very general points \(p_1, \dots, p_r\) on \(\mathbb{P}^2\). Suppose that a curve \(C\) of degree \(d\) is almost-equisingular at \(p_i\) for \(1\leq i \leq r\), and that the loci of log canonical singularities of \(C\) at \(p_i\) are \(0-\)dimensional. Let \(l \in \mathbb{Q}\) denote the log canonical threshold of \(C\) at \(p_i\). Then the inequality \(d \geq \frac{3}{2l}\lfloor \sqrt{r}\rfloor\) holds. Theorem 2. Let \(S\) be a smooth surface, \(C\) an effective divisor on \(S\), and \(p\) a point on \(C\). Let \(m=\text{mult}_p C\) and \(l=\text{lct}(C;p)\). Then, for any \(0 \leq c < 1\), we have \[ \text{colength}\mathcal{J}(S,c\cdot C)_p \geq \lfloor \frac{cm-1}{lm-1}\rfloor \] where \(\mathcal{J}(S,c\cdot C)_P\) is the germ of the multiplier ideal \(\mathcal{J}(S,c\cdot C)\) at \(p\). It is important to notice that we do not know how far the bound of Theorem 1 is from being sharp. However it is substantially better than the known bound for sufficiently large \(r\).
scientific article; zbMATH DE number 5677310
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