On the non existence of some rational 3-folds in ℙ5

In this paper we show that there are no smooth rational 3-folds in ?⁵ (C) which are rational conic bundles, over minimal surfaces, whose generic fibre is embedded as a rational curve of degreeh≥3, (ifh=2 there is a complete classification for these 3-folds as well as for the case of ?¹-bundles). Except for conic bundles, we also give the complete list of rational 3-folds in ?⁵ which are minimal according to Mori’s theory. These are little steps towards the classification of all smooth 3-folds in ?⁵ not of general type.