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Topics in the geometry of projective spaces. Recent work of F. L. Zak. With an addendum by F. L. Zak. (English) Zbl 0564.14007
DMV Seminar, Bd. 4. Basel - Boston - Stuttgart: Birkhäuser Verlag. 52 p. DM 19.80 (1984).
Consider a non singular projective variety \(X\subset {\mathbb{P}}^ m\) of dimension n not contained in any hyperplane. Given a point \(P\in {\mathbb{P}}^ m\setminus X\), projection from P defines a map \(\pi_ p: X\to {\mathbb{P}}^{m-1}\) and it is a classical question to ask whether for generic \(P-\pi_ P\) gives an embedding of X in \({\mathbb{P}}^{m-1}\) or to specify the singularities of \(\pi_ P(X)\). \(\pi_ P\) is always an embedding if \(m>2n+1\), whereas when \(m\leq 2n+1\) this is not true in general. An old result of Severi asserts for instance that if \(n=2\), \(m=5\) only the Veronese surface projects smoothly to \({\mathbb{P}}^ 4\). Recently F. L. Zak proved that if \(3n>2m-4\), \(\pi_ P\) is never an embedding. So it becomes a natural problem to classify varieties on the boundary of the Zak theorem, i.e. \(3n=2m-4\), such that \(\pi_ P\) is an embedding (Severi varieties). - These seminar notes are mainly devoted to a proof of Zak’s very remarkable result: up to projective equivalence there are exactly four Severi varieties: (1) The Veronese surface \(n=2\), \(m=5\); (2) the Segre four-fold \(X={\mathbb{P}}^ 2\times {\mathbb{P}}^ 2\subset {\mathbb{P}}^ 8,\) \(n=4\), \(m=8\); (3) the usual Plücker embedding of the grassmannian G(1,5), \(n=8\), \(m=14\); (4) n\(=16\), \(m=26:\) the only non classical example arising from representation theory of a simply connected algebraic group of type \(E_ 6\). A very suggestive approach to the four Severi varieties has been found by J. Roberts and T. Banchoff: let k denote one of the four division algebras \({\mathbb{R}}, {\mathbb{C}}, {\mathbb{H}}, {\mathbb{Q}}\) and consider \({\mathbb{P}}^ 2(k)\) as a variety over \({\mathbb{R}}\), \(X={\mathbb{P}}^ 2(k)\otimes {\mathbb{C}}\) has a natural embedding into a complete projective space and in this way one gets each of the four Severi varieties.
Zak had never published his proof and his results were known only by private communications. This makes the present paper very useful and interesting. At the end of these notes Zak’s letters about varieties of small codimension are reproduced.
Reviewer: F.Gherardelli

MSC:
14E25 Embeddings in algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14N05 Projective techniques in algebraic geometry