Some remarks on plane curves related to freeness
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910781784719360 |
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| author | Dimca, Alexandru |
| author_facet | Dimca, Alexandru |
| contents | Let $C$ be a reduced complex projective plane curve, and let $d_1$ and $d_2$ be the first two smallest exponents of $C$. For a free curve $C$ of degree $d$, there is a simple formula relating $d,d_1, d_2$ and the total Tjurina number of $C$. Our first result discusses how this result changes when the curve $C$ is no longer free. For a free line arrangement, the Poincaré polynomial coincides with the Betti polynomial $B(t)$ and with the product $P(t)=(1+d_1t)(1+d_2t)$. Our second result shows that for any curve $C$, the difference $P(t)-B(t)$ is a polynomial $a t +bt^2$, with $a$ and $b$ non-negative integers. Moreover $a =0$ or $b=0$ if and only if $C$ is a free line arrangement. Finally we give new bounds for the second exponent $d_2$ of a line arrangement $\mathcal A$, the corresponding lower bound being an improvement of a result by H. Schenck concerning the relation between the maximal exponent of $\mathcal A$ and the maximal multiplicity of points in $\mathcal A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some remarks on plane curves related to freeness Dimca, Alexandru Algebraic Geometry Commutative Algebra Combinatorics Let $C$ be a reduced complex projective plane curve, and let $d_1$ and $d_2$ be the first two smallest exponents of $C$. For a free curve $C$ of degree $d$, there is a simple formula relating $d,d_1, d_2$ and the total Tjurina number of $C$. Our first result discusses how this result changes when the curve $C$ is no longer free. For a free line arrangement, the Poincaré polynomial coincides with the Betti polynomial $B(t)$ and with the product $P(t)=(1+d_1t)(1+d_2t)$. Our second result shows that for any curve $C$, the difference $P(t)-B(t)$ is a polynomial $a t +bt^2$, with $a$ and $b$ non-negative integers. Moreover $a =0$ or $b=0$ if and only if $C$ is a free line arrangement. Finally we give new bounds for the second exponent $d_2$ of a line arrangement $\mathcal A$, the corresponding lower bound being an improvement of a result by H. Schenck concerning the relation between the maximal exponent of $\mathcal A$ and the maximal multiplicity of points in $\mathcal A$. |
| title | Some remarks on plane curves related to freeness |
| topic | Algebraic Geometry Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2501.01807 |