Subcanonicity of codimension two subvarieties
The main concern of the paper is the subcanonical property of some smooth two-codimensional varieties (over the complex field). Actually, the author deals only with rationally numerically subcanonical (shortly r.n.s.) subvarieties, i.e. subvarieties such that the canonical divisor is numerically a rational multiple of the hyperplane section. When \(X\) is a smooth two-codimensional subvariety of \(\mathbb{P}^n\), \(n\in 6\), there are old results by \textit{W. Barth} [Am. J. Math. 92, 951--967 (1970; Zbl 0206.50001)] and \textit{M. E. Larsen} [Invent. Math. 19, 251--260 91973; Zbl 0255.32004)], which this paper recalls in its first section. Then the paper proves that all smooth subvarieties of the Grassmanians \(G(1,4)\) and \(G(1,5)\) are r.n.s. (the proof for \(G(1,5)\) being due to Jorge Caravantes). The paper also shows that the same result is true for a subvariety \(X\) of the smooth quadric \(Q_6\) of \(\mathbb{P}^7\), but under the following assumption: the two genera of the curves obtained by intersecting \(X\) with the two three-dimensional linear spaces of the two families contained in \(Q_6\) must be equal. A counterexample shows that the result cannot be true in general. We must remark that key tools for the proofs are Serre's construction (in order to consider the subvariety as the zero-locus of a set of sections of a suitable vector bundle), and Hodge index theorem. A list of conjectures and problems is given in the last section of the paper.
scientific article; zbMATH DE number 2187069