# 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} }

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.

## 7 Citations

### FREENESS OF HYPERPLANE ARRANGEMENTS AND DIVISORS

- Mathematics
- 2012

This is intended to deliver as a lecture note “Arrangements in Pyrénées”. The main theme is free arrangements. We are discussing relations of freeness and splitting problems of vector bundles,…

### 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…

### Freeness of hyperplane arrangements and related topics

- Mathematics
- 2012

This is the expanded notes of the lecture by the author in "Arrangements in Pyrenees", June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques…

### Freeness and near freeness are combinatorial for line arrangements in small degrees

- Mathematics
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We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is…

### Characteristic polynomials, $\eta$-complexes and freeness of tame arrangements

- Mathematics
- 2011

We compare each coefficient of the reduced characteristic polynomial of a simple arrangement and that of its Ziegler restriction. As a consequence we can show that the former is not less than the…

### Arrangements, multiderivations, and adjoint quotient map of type ADE

- Mathematics
- 2010

The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham…

### Roots of characteristic polynomials and intersection points of line arrangements

- Mathematics
- 2013

We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we…

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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…

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