  The pentagram map, introduced by R. Schwartz, is a birational map on the
configuration space of polygons in the projective plane.
  We study the singularities of the iterates of the pentagram map. We show that
a "typical" singularity disappears after a finite number of iterations, a
confinement phenomenon first discovered by Schwartz. We provide a method to
bypass such a singular patch by directly constructing the first subsequent
iterate that is well-defined on the singular locus under consideration. The key
ingredient of this construction is the notion of a decorated (twisted) polygon,
and the extension of the pentagram map to the corresponding decorated
configuration space.
