Moduli spaces of polygons have been, since the ’90s, a widely studied example of K¨ahler reduction. The hyperpolygons spaces are the non-compact hyperK¨ahler quotient analogue to polygons spaces.

In the first part of the talk we will introduce the moduli space M_r of polygons (in the Euclidean space) with edges of length r_1, . . . , r_n. These spaces arise by symplectic reduction performed at the r-level set, r = (r_1, . . . , r_n) \in R^{n}_{+} being the lengths vector. We analyze the birational map that encodes the changes in M_r when r crosses a wall, showing that for r and r′ in different chambers the flip between M_r and M_r′ can be described in terms of moduli spaces of polygons of lower dimension. The second part of the talk will move in the hyperK¨ahler setting: the hyperpolygons space X_r carries a natural S^{1}-action and the core L_r of X_r is defined to be the union of the flow down sets from the connected components of the fixed points set (X_r)S^{1}. We will illustrate some known results on hyperpolygons spaces and their core, and discuss their relation with polygon spaces and with moduli spaces of parabolic Higgs bundles (on-going joint work with Leonor Godinho).