  The pentagram map was introduced by R. Schwartz in 1992 for convex planar
polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved
Liouville integrability of the pentagram map for generic monodromies by
providing a Poisson structure and the sufficient number of integrals in
involution on the space of twisted polygons. In this paper we prove
algebraic-geometric integrability for any monodromy, i.e., for both twisted and
closed polygons. For that purpose we show that the pentagram map can be written
as a discrete zero-curvature equation with a spectral parameter, study the
corresponding spectral curve, and the dynamics on its Jacobian. We also prove
that on the symplectic leaves Poisson brackets discovered for twisted polygons
coincide with the symplectic structure obtained from Krichever-Phong's
universal formula.
