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Abstract

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 its further application to the exponents. In fact, exponents of $2$-dimensional multiarrangements are calculated by verifying whether the corresponding matrices have full rank. As the main result, we introduce a new class of $2$-dimensional arrangements with an explicit sequence of multiplicities for which the exponents are closest. We also provide an alternative proof for some known results on exponents.