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Plus-one generated and next to free arrangements of hyperplanes
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We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show that a free arrangement is in fact surrounded by free or strictly plus-one generated arrangements. We also give several applications.
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Combinatorially Determined Zeroes of Bernstein--Sato Ideals for Tame and Free Arrangements
For tame and free hyperplane arrangements, the zero loci of Bernstein-Sato ideals and the roots in [-1,0) of Bernstein-Sato polynomials are determined by the intersection lattice, with explicit combinatorial formulas.
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