The moment polytope of the abelian polygon space

Abstract

The moduli space of chains in the plane with generic side lengths is a smooth, closed manifold. Interestingly, this manifold is also the fixed point set of the complex conjugation on a toric variety, known as the abelian polygon space. Hence, affording the structure of a real toric variety. In this paper we show that the moment polytope of the moduli space of chains can be determined by the combinatorial data, called the short code of the length vector. We also classify chain spaces which are aspherical using a result of Davis, Januszkiewicz and Scott.