# Some Special Properties of the Adjunction Theory for \(3\)-Folds in \(\mathbb P^{5}\)

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*Mauro C. Beltrametti; Michael Schneider; Andrew J. Sommese*

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

#### Table of Contents

# Table of Contents

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

- Contents vii8 free
- Introduction 110 free
- Chapter 0. Background material 312 free
- Chapter 1. The second reduction for n–folds in P[sup(2n-1)] 1120
- Chapter 2. General formulae for threefolds in P[sup(5)] 1928
- Chapter 3. Nefness and bigness of K[sub(x)+ 2K 2534
- Chapter 4. Ampleness of K[sub(x)+ 2K 3342
- Chapter 5. Nefness and bigness of K[sub(x)+ K 3948
- Chapter 6. Invariants for threefolds in P[sub(5)] up to degree 12 5160
- References 6170