On the module of derivations of a line arrangement
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910954508255232 |
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| author | Dimca, Alexandru |
| author_facet | Dimca, Alexandru |
| contents | To each multiple point $p$ in a line arrangement $ \mathcal A$ in the complex projective plane we associate a local derivation $\tilde D_p \in D_0( \mathcal A)$. We show first that these derivations span the graded module of derivations $D_0( \mathcal A)$ in all degrees $\geq d -3$, where $d$ is the number of lines in $ \mathcal A$, see Theorem 1.4 and Theorem 1.6. Then, to each local derivation $\tilde D_p \in D_0( \mathcal A)$ we associate a polynomial $g_p$ which seems to play a key role in the characterization of the freeness of $ \mathcal A$, see Theorem 1.10, as well as in the study of the position of the multiple points of $ \mathcal A$ with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the module of derivations of a line arrangement Dimca, Alexandru Algebraic Geometry Combinatorics To each multiple point $p$ in a line arrangement $ \mathcal A$ in the complex projective plane we associate a local derivation $\tilde D_p \in D_0( \mathcal A)$. We show first that these derivations span the graded module of derivations $D_0( \mathcal A)$ in all degrees $\geq d -3$, where $d$ is the number of lines in $ \mathcal A$, see Theorem 1.4 and Theorem 1.6. Then, to each local derivation $\tilde D_p \in D_0( \mathcal A)$ we associate a polynomial $g_p$ which seems to play a key role in the characterization of the freeness of $ \mathcal A$, see Theorem 1.10, as well as in the study of the position of the multiple points of $ \mathcal A$ with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component. |
| title | On the module of derivations of a line arrangement |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2503.01624 |