Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$

This work studies the adjunction theory of smooth $3$-folds in $\mathbb P^5$. Because of the many special restrictions on such $3$-folds, the structure of the adjunction theoretic reductions are especially simple, e.g. the $3$-fold equals its first reduction, the second reduction is smooth except possibly for a few explicit low degrees, and the formulae relating the projective invariants of the given $3$-fold with the invariants of its second reduction are very explicit. Tables summarizing the classification of such $3$-folds up to degree $12$ are included. Many of the general results are shown to hold for smooth projective $n$-folds embedded in $\mathbb P^N$ with $N \leq 2n-1$.