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Register a new userDeletion theorem and combinatorics of hyperplane arrangements
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Authors
Takuro Abe
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We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
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In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Teraos celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Teraos conjecture is true in this class. As an application, we can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.
In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended Catalan arrangements are given.
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.
There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. We expand it as resolving the complexity issue expected by Mnevs universality theorem and conduct combinatorializing so the theory over fields becomes realization of our combinatorial theory. A main theorem is that for n less than or equal to 9 a specific and general enough kind of matroid tilings in the hypersimplex Delta(3,n) extend to matroid subdivisions of Delta(3,n) with the bound n=9 sharp. As a straightforward application to realizable cases, we solve an open problem in algebraic geometry proposed in 2008.
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