Normal bicanonical and tricanonical threefolds

The first author to construct a non-normal bicanonical threefold in P 4 was L. Godeaux in 1936 [5]. This threefold has degree 8. In the first part of the present paper, starting from a normal threefold of general type where q1 = q2 = pg = 0, P2 = P3 = 5, of degree 6 (cf. [11]), we construct Godeaux’s example and two examples of tricanonical threefolds in P 4 . One of the tricanonical threefolds is normal. In the second part of the paper, we construct (starting from the beginning) a normal bicanonical threefold of degree 8 that has the birational invariants given by q1 = q2 = pg = 0 and P2 = 5. No other examples of bicanonical and tricanonical threefolds in P 4 are known.