Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918402646343680 |
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| author | Maehara, Shota |
| author_facet | Maehara, Shota |
| contents | In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of $2$-dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of $2$-dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of $B_2$-exponents, and we provide an alternative proof for some known results on the exponents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices Maehara, Shota Combinatorics In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of $2$-dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of $2$-dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of $B_2$-exponents, and we provide an alternative proof for some known results on the exponents. |
| title | Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.16569 |