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Published online by Cambridge University Press: 24 October 2008
The present paper is a sequel to a previous study (7) of the completely regular threefolds which possess anticanonical systems, i.e. for which the virtual canonical system, reversed in sign, is effective of positive order. On any such threefold the process of successive adjunction, applied to any linear system of surfaces, must terminate; we have thus to deal with a special case of the adjunction problem for the regular threefolds. By making certain simplifying hypotheses (such as irreducibility and absence of base elements) concerning the anticanonical systems, one can classify the threefolds in broad outline and show that, provided the anticanonical systems are sufficiently ample, the corresponding threefolds are either unirational or birational.