Deletion-restriction in toric arrangements

Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric arrangements for which the cohomology algebra of the complement is generated in degree $1$. We also show that for these arrangements the complement is formal in the sense of Sullivan.Comment: v2: typos fixed, 11 pages. Accepted for publication in Journal of Ramanujan Mathematical Societ

Orbit closures of representations of source-sink Dynkin quivers

We use the geometric technique, developed by Weyman, to calculate the resolution of orbit closures of representations of Dynkin quivers with every vertex being source or sink. We use this resolution to derive the normality of such orbit closures. As a consequence we obtain the normality of certain orbit closures of type E.Comment: 11 pages, 3 figure

Face counting formula for toric arrangements defined by root systems

A toric arrangement is a finite collection of codimension-1 subtori in a torus. The intersections of these subtori stratify the ambient torus into faces of various dimensions. The incidence relations among these faces give rise to many interesting combinatorial problems. One such problem is to obtain a closed-form formula for the number of faces in terms of the intrinsic arrangement data. In this paper we focus on toric arrangements defined by crystallographic root systems. Such an arrangement is equipped with an action of the associated Weyl group. The main result is a formula that expresses the face numbers in terms of a sum of indices of certain subgroups of this Weyl group. Keywords: Toric arrangements, Face enumerations, f-vector, Affine Weyl group