The manifold of polygons degenerated to segments

In this paper we study the space L (n) of n-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of Cⁿ, and describe its topology in terms of the manifold M (n) of n-gons degenerated to segments and with the first vertex at 0. We show that M (n) and L (n) contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of L (n) by the diagonal action of the affine complex group and the re-enumeration of the vertices is described.