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On triple covers of \(\mathbb{P}^ n\) as very ample divisors. (English) Zbl 0841.14003
Ciliberto, Ciro (ed.) et al., Classification of algebraic varieties. Algebraic geometry conference on classification of algebraic varieties, May 22-30, 1992, University of L’Aquila, L’Aquila, Italy. Providence, RI: American Mathematical Society. Contemp. Math. 162, 277-292 (1994).
The authors’ aim is to classify the pairs \((X,L)\) where \(X\) is a projective manifold and \(L\) a very ample line bundle on \(X\) such that the linear system \(|L |\) contains a smooth divisor \(A\) which admits a finite morphism \(\pi : A \to \mathbb{P}^n\) of degree 3. One assumes \(n \geq 2\). The following examples are given:
\((\text{a}_n)\) \((\mathbb{P}^{n+1}, {\mathcal O}_{\mathbb{P}^{n + 1}} (3))\);
\((\text{b}_n)\) a cubic hypersurface in \(\mathbb{P}^{n + 2}\) with hyperplane section;
\((\text{c}_n)\) \((X,3L)\) where \((X,L)\) is del Pezzo, i.e. \(- K_X = nL\) is ample, with \(L^{n + 1} = 1\).
The authors show that for \(n \geq 4\) the above examples are the only ones. In case \(n = 3\) the examples are the only ones if one makes the additional assumption that the cover \(\pi : A \to \mathbb{P}^3\) is cyclic. The case \(n = 2\) is more complicated; it is studied under the additional requirement \(K_A = \pi^* {\mathcal O}_{\mathbb{P}^2} (k)\) (for some integer \(k)\) and some results are given.
For the entire collection see [Zbl 0791.00020].

14C20 Divisors, linear systems, invertible sheaves
14E20 Coverings in algebraic geometry
14N05 Projective techniques in algebraic geometry