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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].

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14E20 Coverings in algebraic geometry 14N05 Projective techniques in algebraic geometry
##### Keywords:
triple covers; very ample line bundle; linear system