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 \(N \leq 2n-1\).

Readership

Research mathematicians, researchers in algebraic geometry.

Table of Contents

• Introduction
• Background material
• The second reduction for \(n\)-folds in \(\mathbb P^{2n-1}\)
• General formulae for threefolds in \(\mathbb P^5\)
• Nefness and bigness of \(K_X+2\mathcal K\)
• Ampleness of \(K_X+2\mathcal K\)
• Nefness and bigness of \(K_X+\mathcal K\)
• Invariants for threefolds in \(\mathbb P^5\) up to degree \(12\)
• References