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Projective normality of varieties of small degree. (English) Zbl 0920.14012

The classification of projective smooth complex varieties of small degree was accomplished by Ionescu, Okonek, Alexander and very recently completed by Abo, Decker and Sasakura. Our interest is in the projective normality of these varieties. A projective variety \(X\subset\mathbb{P}^N\) is \(k\)-normal if \(\tau_k:H^0(\mathbb{P}^N,{\mathcal O}_{\mathbb{P}^N}(k))\to H^0(X,{\mathcal O}_X(k))\) is surjective. \(X\) is linearly normal if it is 1-normal and it is projectively normal (p.n.), if \(\tau_k\) is surjective for all \(k\geq 1\). It is known that all linearly normal varieties of degree \(d\leq 5\) are p.n. [see e.g. A. Ohbuchi, Pac. J. Math. 144, No. 2, 313-325 (1990; Zbl 0759.14005)]. In his classification papers in: Algebraic geometry, Proc. int. Conf., Bucharest 1982, Let. Notes Math. 1056, 142-186 (1984; Zbl 0542.14024) and in: Algebraic geometry, Proc, int. Conf., L’Aquila 1988, Lect. Notes Math. 1417, 138-154 (1990; Zbl 0709.14023), P. Ionescu already established the projective normality and more generally the arithmetical normality in some cases for \(6\leq d\leq 8\). In the work under review the projective normality of varieties of degree \(6\leq d\leq 8\) is thoroughly investigated. The complete list of non projectively normal such manifolds is given; all of them are shown to be not 2-normal.

MSC:

14J10 Families, moduli, classification: algebraic theory
14F45 Topological properties in algebraic geometry
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References:

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