On smooth rational threefolds of P-5 with rational non-special hyperplane section

It is known that the smooth rational threefolds of P-5 having a rational no n-special surface of P-4 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 P-3, looking in each case for an explicit description of a bir ational map from P-3 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 c ompletely 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 brewing-up of P-3 along B. If d = 7, i.e., if X is a Palatini scroll , we prove that, conversely, a similar projection never exists.