INTRODUCTION

We report on some recent progress in the classification of smooth projective varieties with small invariants. This progress is mainly due to the finer study of the adjunction mapping by Reider, Sommese and Van de Ven [SoI], [VDV], [Rei], [SV]. Adjunction theory is a powerful tool for determining the type of a given variety. Classically, the adjunction process was introduced by Castelnuovo and Enriques [CE] to study curves on ruled surfaces. The Italian geometers around the turn of the century also started the classification of smooth surfaces in ℙ4 of low degree. Further classification results are due to Roth [Rol], who uses the adjunction mapping to get surfaces with smaller invariants already known to him (compare [Ro2] for adjunction theory on 3-folds). Nowadays, through the effort of several mathematicians, a complete classification of smooth surfaces in ℙ4 and smooth 3-folds in ℙ5 has been worked out up to degree 10 and 11 respectively. Moreover, in the 3-fold case the classification is almost complete in degree 12. For references see section 7.

One motivation to study these varieties comes from Hartshorne's conjecture [Hal]. In the case of codimension 2 this suggests that already smooth 4- folds in ℙ6 should be complete intersections.