The homotopy type of elliptic arrangements

We give combinatorial models for the homotopy type of complements elliptic arrangements (i.e., certain sets abelian subvarieties in a product curves). presentation fundamental group such spaces and, as an application, we treat case ordered configuration curves. Our are finite polyhedral CW complexes, and our tools choice acyclic categories (small without loops). As stepping stone, characterization which arise face complexes.