Primitivity in twisted homogeneous coordinate rings

Given a projective k-scheme X, an automorphism sigma of X and an invertible sheaf L on X, one can form the twisted homogeneous coordinate ring bigoplus_{nge 0} H0(X,L_n), where L_n=Lotimes L^{sigma}otimes cdotsotimes L^{sigma^{n-1}. We study the question of primitivity of such rings. A ring R is primitive if it has a maximal left ideal which does not contain a nonzero two sided ideal. We show in many cases that primitivity of twisted homogeneous coordinate rings is equivalent to the quotient division ring having trivial centre and to the ring having finitely many height one primes. This gives a Dixmier-Moeglin correspondence for many classes of twisted homogeneous coordinate rings.