We answer a question by Jonathan Wahl, giving examples of regular surfaces (so that the canonical ring is Gorenstein) with the following properties:
(1) the canonical divisor KS ≡ 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 KX ℚ-Cartier which are not ℚ-Gorenstein, since their index one cover is not Cohen-Macaulay.
Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday
The situation that we consider in this paper is the following: L is an ample divisor on a complex projective manifold X of complex dimension n, and we assume that L is subcanonical, i.e., there exists an integer h such that we have the linear equivalence KX ≡ hL, where h ≠ 0. There are then two cases: h < 0 and X is a Fano manifold, or h > 0 and X is a manifold with ample canonical divisor (in particular X is of general type). Assume that X is a Fano manifold and that −KX = rL, with r > 0: then, by Kodaira vanishing,
Hj(mL) := Hj(OX(mL)) = 0, ∀m ∈ ℤ,∀ 1 ≤ j ≤ n − 1.
For m < 0 this follows from Kodaira vanishing (and holds for j ≥ 1), while for m ≥ 0 Serre duality gives hj(mL) = hn − j(K − mL) = hn−j((−r −m)L) = 0. At the other extreme, if KX is ample and KX ≡ rL (thus r > 0), by the same argument we get vanishing outside of the interval
0 ≤ m ≤ r.