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

TL;DR

This survey reviews the classification of smooth surfaces in P^4 and 3-folds in P^5, highlighting results from adjunction theory, example constructions via syzygies, and known families of non-general type varieties.

Contribution

It compiles and explains classification results, construction methods, and known families of smooth varieties in P^4 and P^5, emphasizing syzygy techniques.

Findings

List of all known families of non-general type surfaces in P^4.

List of all known families of non-general type 3-folds in P^5.

Discussion of construction methods via syzygies.

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.