On Surfaces in P

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 [So1], [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 P of low degree. Further classification results are due to Roth [Ro1], 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 P and smooth 3-folds in P has been worked out up to degree 10 and 11 resp. 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 [Ha1]. In the case of codimension 2 this suggests that already smooth 4-folds in P should be complete intersections. Another motivation originates from two mutually corresponding finiteness results. Ellingsrud and Peskine [EP] proved that there are only finitely many families of smooth surfaces in P which are not of general type. However, the question of an exact degree bound d0 is still open. By [BF] d0 ≤ 105. Examples are known only up to degree 15 and one actually believes that d0 = 15. The analogous finiteness result holds for 3-folds in P [BOSS1]. In this case one expects a much higher degree bound. Nevertheless examples had been known so far only up to degree 14 [Ch3]. In this note we present, among other things, three new smooth 3-folds in P of degree 13, 17 and 18 resp. How to construct examples ?