## Projective manifolds with hyperplane sections being five-sheeted covers of projective space.(English)Zbl 1119.14008

Let $$X$$ be a smooth complex projective variety of dimension $$n+1$$ and let $$L$$ be a very ample line bundle on $$X$$. Suppose that $$| L|$$ contains a smooth member $$A$$ endowed with a branched covering $$\pi:A \to \mathbb P^n$$ of degree $$d$$. Classical results on surfaces with hyperelliptic curves as hyperplane sections and their revision made in the 1980’s called the attention to the problem of classifying pairs $$(X,L)$$ as above. The problem has been considered by many authors according to the following values of $$(n,d)$$. For $$(1,2)$$ by A. J. Sommese and A. Van de Ven [Math. Ann. 278, 593–603 (1987; Zbl 0655.14001)]; for $$(1,3)$$ by Serrano (unpublished) and by M. L. Fania [Manuscr. Math. 68, 17–34 (1990; Zbl 0729.14028)]; for $$(n,2), (n,3)$$ by M. Palleschi, A. J. Sommese and the reviewer [Nagoya Math. J. 137, 1–32 (1995; Zbl 0820.14005); in: Classification of algebraic varieties. Conf. 1992, L’Aquila, Italy. Contemp. Math. 162, 277–292 (1994; Zbl 0841.14003)]; for $$(n,d)$$ with $$n > d=4,5$$ by the reviewer in [Papers in honor of Giovanni Melzi. Milano: Univ. Cattolica del Sacro Cuore. Sci. Mat. 11, 231–248 (1994; Zbl 0831.14023)].
In the paper under review, the author focuses on case $$(n,5)$$ with $$n \geq 6$$ again, providing a complete classification result. In particular, one case, described only in terms of numerical invariants and pointed out as possible by the reviewer, is settled positively. In addition to the four pairs $$(X,L)$$ already known, the author proves that $$X$$ is a smooth weighted hypersurface $$W \subset \mathbb P(5,4,1^{n+1})$$ of degree $$20$$ with $$L=\mathcal O_W(5)$$, which in fact turns out to be very ample. Recently the author considered also the case $$(n,4)$$ with $$n \geq 5$$ [Proc. Japan Acad., Ser. A Math. Sci. 82, 8–13 (2006; Zbl 1106.14002)].

### MSC:

 14C20 Divisors, linear systems, invertible sheaves 14J40 $$n$$-folds ($$n>4$$) 14H30 Coverings of curves, fundamental group 14N30 Adjunction problems
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### References:

 [1] G. Castelnuovo, Sulle superficie algebriche le cui sezioni piane sone curve iperellittiche, Rend. Circ. Mat. Palermo, 4 (1890), 73-88. · JFM 22.0788.01 [2] M. L. Fania, Trigonal hyperplane sections of projective surfaces, Manuscr. Math., 68 (1990), 17-34. · Zbl 0729.14028 [3] T. Fujita, Vector bundles on ample divisors, J. Math. Soc. Japan, 33 (1981), 405-414. · Zbl 0475.14014 [4] T. Fujita, On the structure of polarized manifolds of total deficiency one, III, J. Math. Soc. Japan, 36 (1984), 75-89. · Zbl 0541.14036 [5] T. Fujita, Classification theories of polarized varieties, London Math. Soc. Lecture Note Ser., 155 , Cambridge Univ. Press, Cambridge 1990. · Zbl 0743.14004 [6] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52 , Springer, 1977. · Zbl 0367.14001 [7] P. Ionescu, Embedded projective varieties of small invariants, In: Algebraic geometry, Bucharest, 1982, Lecture Notes in Math., 1056 , Springer, 1984, pp.,142-186. · Zbl 0542.14024 [8] A. Laface, A very ampleness result, Matematiche, 52 (1997), 431-442. · Zbl 0955.14004 [9] A. Lanteri, Small degree covers of $$\bm{P}^{n}$$ and hyperplane sections, In: Writings in honor of Giovanni Melzi, Sci. Mat., 11 , Vita e Pensiero, Milan, 1994, pp.,231-248. · Zbl 0831.14023 [10] A. Lanteri, M. Palleschi and A. J. Sommese, Double covers of $$\bm{P}^{n}$$ as very ample divisors, Nagoya Math. J., 137 (1995), 1-32. · Zbl 0820.14005 [11] A. Lanteri, M. Palleschi and A. J. Sommese, On triple covers of $$\bm{P}^{n}$$ as very ample divisors, In: Classification of algebraic varieties, L’Aquila 1992, Contemp. Math., 162 , Amer. Math. Soc., Providence, RI. 1994, pp.,277-292. · Zbl 0841.14003 [12] R. Lazarsfeld, A Barth-type theorem for branched coverings of projective space, Math. Ann., 249 (1980), 153-162. · Zbl 0434.32013 [13] S. Mori, On a generalization of complete intersections, J. Math. Kyoto Univ., 15 (1975), 619-646. · Zbl 0332.14019 [14] F. Serrano, The adjunction mapping and hyperelliptic divisors on a surface, J. Reine Angew. Math., 381 (1987), 90-109. · Zbl 0618.14001 [15] A. J. Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann., 278 (1987), 593-603. · Zbl 0655.14001 [16] K. -i. Watanabe, Some remarks concerning Demazure’s construction of normal graded rings, Nagoya Math. J., 83 (1981), 203-211. · Zbl 0518.13003
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