The addition-deletion theorems for hyperplane arrangements, which were originally shown by Terao [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 293-320.], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements, including supersolvable arrangements and the Coxeter arrangement of type A3, to construct free and non-free multiarrangements.
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