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

Abstract

It is known that the smooth rational threefolds of P^5 having a rational non-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 fosr an explicit description of a birational 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 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 P^3 along B. If d=7, i.e. if X is a Palatini scroll, we prove that, conversely, a similar projection never exists.