On surfaces in P^4 and 3-folds in P^5

Abstract

This is a survey on the classification of smooth surfaces in P^4 and smooth 3-folds in P^5. We recall the corresponding results arising from adjunction theory and explain how to construct examples via syzygies. We discuss some examples in detail and list all families of smooth non-general type surfaces in P^4 and 3-folds in P^5 known to us.