On threefolds with the smallest nontrivial monodromy group
Using an adjunction-theoretic result due to \textit{A. J. Sommese} [J. Reine Angew. Math. 402, 211--220 (1989; Zbl 0675.14005)] together with a proposition from [Séminaire de géométrie algébrique du Bois-Marie 1967--1969. Groupes de monodromie en géométrie algébrique (SGA 7 II) par P. Deligne et N. Katz. Exposés X à XXII. Cham: Springer (1973; Zbl 0258.00005)], the author obtains a complete list of smooth threefolds for which the monodromy group, acting on the second cohomology group of its smooth hyperplane section, is \(\mathbb{Z}/2\mathbb{Z}\). The list includes the quadric \(Q\subset \mathbb{P}^4\), the Veronese variety or its isomorphic projection, the Segre variety and the blowup of \(\mathbb{P}^3\) at a point, embedded in \(\mathbb{P}^8\) by the complete linear system \(|2H-E|\), where \(H\) is the preimage of a plane in \(\mathbb{P}^3\) and \(E\) is the exceptional divisor, or an isomorphic projection of this variety. The possibility of such a classification was announced by \textit{F. L. Zak} [Lect. Notes Math. 1479, 273--280 (1991; Zbl 0793.14026)].
scientific article; zbMATH DE number 7917519
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