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Severi varieties. (Russian) Zbl 0577.14031

It is a classical result that any smooth n-dimensional variety \(X^ n\subset P^ N\) can be projected isomorphically into \(P^{N-1}\) if \(N\geq 2n+2\). Of course, it is natural to suppose \(X^ n\not\subset P^{N-1}\) for any hyperplane \(P^{N-1}\subset P^ N\). In the converse direction Harthshorne conjectured that if \(X^ n\subset P^ N\), \(X^ n\not\subset P^{N-1}\) and \(X^ n\) can be projected isomorphically into \(P^{N-1}\), then \(N\geq 3n/2+2\). This has been proved by the author in Mat. Sb., Nov. Ser., 116(158), 593-602 (1981; Zbl 0484.14016). In this connection, of particular interest are the varieties \(X^ n\subset P^ N\) \((N\geq 3n/2+2)\), for which there exists an isomorphic projection into \(P^{3n/2+1}\). The author calls them Severi varieties and presents in this paper their full classification over an algebraically closed field of characteristic zero. Namely, there exist exactly four Severi varieties, one in each of the dimensions \(n=2, 4, 8\) and 16 respectively. These turn out to be the Veronese surface in \(P^ 5\), the Segre variety \(P^ 2\times P^ 2\subset P^ 8\), the Grassmannian \(G(5,1)\subset P^{14}\), and the 16-dimensional variety in \(P^{26}\), recently discovered by Lazarsfeld [see R. Lazarsfeld, ”An example of 16- dimensional projective variety with a 25-dimensional secant variety”, Math. Letters 7, 1-4 (1981)]. In particular, the estimate in the Hartshorne conjecture can be strengthed for \(n>16\).
Reviewer: V.Iliev

MSC:

14J40 \(n\)-folds (\(n>4\))

Citations:

Zbl 0484.14016
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