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The paper answers a question by Jonathan Wahl,giving examples of regularsurfaces S (so their canonical ring is a Gorenstein graded ring) having thefollowing properties:  1) their canonical divisor K_S = rL is a positive multiple of an ampledivisor 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 ofCohen-Macaulay singularities with K_X Q -Cartier which are not Q -Gorenstein,since their index one cover is not Cohen- Macaulay.

Subcanonical Graded Rings Which Are Not Cohen-Macaulay, with Appendix: A non-Q-Gorenstein Cohen-Macaulay cone X with K_X Q-Cartier