• Corpus ID: 117949112

# Exponents of 2-multiarrangements and freeness of 3-arrangements

@article{Abe2010ExponentsO2,
title={Exponents of 2-multiarrangements and freeness of 3-arrangements},
author={Takuro Abe},
journal={arXiv: Commutative Algebra},
year={2010}
}
• T. Abe
• Published 28 May 2010
• Mathematics
• arXiv: Commutative Algebra
We give the upper bound of differences of exponents for balanced 2-multiarrangements in terms of the cardinality of hyperplanes. Also, we give a shift isomorphism of 2-multiarrangements like Coxeter arrangements when the difference of exponents is maximum. As an application, a sufficient numerical and combinatorial condition for 3-arrangements to be free is given.

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## References

SHOWING 1-10 OF 19 REFERENCES

### On the Exponents of 2-Multiarrangements

. In this paper we study the exponents of 2-multiarrangements. More precisely, we compose a basis for D( A ,k) in the case where A consists of three lines using Q -polynomials (cid:1) Xλ (cid:2) .

### Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula

We define an n-arrangement as a finite family of hyperplanes through the origin in C "+1. In [11] and [12] we studied the free arrangement and defined its structure sequence (their definitions will

### The Stability of the Family of B 2-Type Arrangements

We introduce the family of B 2-type arrangements as a generalization of the classical Coxeter arrangement of type B 2 and consider the stability and the freeness of it. We show the freeness and

### The primitive derivation and freeness of multi-Coxeter arrangements

We will prove the freeness of multi-Coxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito's

### Exponents of 2-multiarrangements and multiplicity lattices

• Mathematics
• 2012
We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation

### Characterization of a free arrangement and conjecture of Edelman and Reiner

We consider a hyperplane arrangement in a vector space of dimension four or higher. In this case, the freeness of the arrangement is characterized by properties around a fixed hyperplane. As an

### On the Freeness of 3‐Arrangements

Hyperplane arrangements in a three‐dimensional vector space are considered in this paper. A characterization of the freeness of such an arrangement is given in terms of the characteristic polynomial

### A generalized logarithmic module and duality of Coxeter multiarrangements

We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a

### The Euler multiplicity and addition–deletion theorems for multiarrangements

• Mathematics
• 2006
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