  The paper answers a question by Jonathan Wahl,giving examples of regular
surfaces S (so their canonical ring is a Gorenstein graded ring) having the
following properties:
  1) their canonical divisor K_S = rL is a positive multiple of an ample
divisor L
  2) the graded ring R := R (X,L ) associated to L is not Cohen-Macaulay.
  In the appendix Wahl shows how these examples lead to the existence of
Cohen-Macaulay singularities with K_X Q -Cartier which are not Q -Gorenstein,
since their index one cover is not Cohen- Macaulay.
