Abstract
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.
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