  A rational map whose source and image are projectively embedded varieties has
an {\em Arithmetically Cohen-Macaulay graph} if the Rees algebra of one (hence
any) of its base ideals is a Cohen-Macaulay ring. If the map is birational onto
the image one considers how this property forces an upper bound on the degree
of a representative of the map. In the plane case a complete description is
given of the Cremona maps with Cohen-Macaulay graph, while in arbitrary
dimension $n$ it is shown that a Cremona map with Cohen-Macaulay graph has
degree at most $n^2$.
