Free reflection multiarrangements and quasi-invariants

To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system $BC_N$.Comment: 26 pages; small change

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement

Numerical invariants and moduli spaces for line arrangements

Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.Comment: v3: A new proof of a result due to Tohaneanu, giving the classification of line arrangements with a Jacobian syzygy of minimal degree 2 is given in Theorem 4.11. Some other minor change

Topological Criteria for k−k-Formal Arrangements

We prove a criterion for $k-$formality of arrangements, using a complex constructed from vector spaces introduced in \cite{bt}. As an application, we give a simple description of $k-$formality of graphic arrangements: Let $G$ be a connected graph with no loops or multiple edges. Let $\Delta$ be the flag (clique) complex of $G$ and let $H_{\bullet}(\Delta)$ be the homology of the chain complex of $\Delta$. If $\mathcal A_G$ is the graphic arrangement associated to $G$, we will show that $\mathcal A_G$ is $k-$formal if and only if $H_i(\Delta)=0$ for every $i=1,...,k-1$.Comment: 9 pages, 1 figur