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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.

14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
14M06 Linkage
Full Text: DOI EuDML
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