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

Abstract

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 P 3 , looking in each case for an explicit description of a birational map from P3 t o 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.

Weakly-exceptional singularities in higher dimensions

“…If X is smooth, then it follows from [41, Proposition 1.7] that X is a so-called smooth Bordiga scroll, which is a projectivization of a two-dimensional stable vector bundle E on P 2 such that c 1 (E) = 0 and c 2 (E) = 0. Smooth Bordiga scrolls have been studied in [41], [43] and [38]. One can show that smooth Bordiga scrolls are weak Fano threefolds, i.e.…”