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Threefolds of degree 9 and 10 in $${\mathbb{P}}^ 5$$. (English) Zbl 0723.14033
Let X be a smooth connected variety of dimension $$n\geq 2$$ with canonical bundle $$K_ X$$. Let L be a very ample line bundle on X. If the global sections of $$K_ X+(n-1)L$$ span it, then they determine a map $$\Phi: X\to {\mathbb{P}}^ m$$, called the adjunction map. A detailed study of adjunction maps has been done by A. J. Sommese (and also others) in a series of papers [e.g. in Complex analysis and algebraic geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 175-213 (1986; Zbl 0601.14029)].
This paper carries it further $$(n=3)$$ to complete the classification of smooth threefolds X of degree 9 and 10 in $${\mathbb{P}}^ 5$$. For such an X, it is shown that dim($$\Phi$$ (X))$$\geq 2$$. If $$\dim(\Phi(X))\neq 3$$, then X is a $${\mathbb{P}}^ 1$$-bundle over a minimal $$K_ 3$$ surface, degree$$(X)=9$$, For $$\dim(\Phi(X))=3$$, the classification uses minimal reduction $$(X_ 0,L_ 0)$$ of (X,L); $$(X_ 0,L_ 0)$$ is shown to be either of log general type or a conic bundle over a surface. For each class, examples are constructed using liaison. They turn out to be unique examples with given invariants.
The classification of smooth threefolds of degree $$\leq 8$$ in $${\mathbb{P}}^ 5$$ has been done by C. Okonek [Manuscr. Math. 38, 175-199 (1982; Zbl 0534.14023)] and Ionescu independently.

##### MSC:
 14J30 $$3$$-folds 14J10 Families, moduli, classification: algebraic theory 14M06 Linkage
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##### References:
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