On smooth rational threefolds of P5 with rational non—special hyperplane section

It is known that the smooth rational threefolds of P⁵ having a rational non—special surface of P⁴ as general hyperplane section have degree d = 3, …, 7. We study such threefolds X from the point of view of linear systems of surfaces in P³, looking in each case for an explicit description of a birational map from P³ to X. For d = 3, … 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing—down of the blowing—up of P³ along B. If d = 7, i.e., if X is a Palatini scroll, we prove that, conversely, a similar projection never exists.