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Linear systems of hyperplane sections on varieties of low codimension. (English. Russian original) Zbl 0577.14032
Funct. Anal. Appl. 19, 165-173 (1985); translation from Funkts. Anal. Prilozh. 19, No. 3, 1-10 (1985).
Let $$X^ n\subset {\mathbb{P}}^ N$$ be a non-degenerate smooth projective variety over an algebraically closed field, and suppose that X can be isomorphically projected to $${\mathbb{P}}^{2n+1-\delta}$$, where $$\delta >0$$. Then it turns out that $$N\leq f(Ent(n/\delta))$$, where f is a quadratic function whose explicit form is computed in the paper. In other words, if $$X^ n\subset {\mathbb{P}}^ r$$ has ”small codimension”, i.e. $$r\leq 2n$$, then $$h^ 0(X,{\mathcal O}_ X(1))\leq f(Ent(n/2n+1-r))+1.$$
The above result follows from a study of higher secant varieties; in particular, if $$S^ kX$$ denotes the variety of k-dimensional secant subspaces of X (such that $$S^ 0X=X$$ and $$S^ 1X=SX$$ is the usual secant variety), then it is shown that $$S^{Ent(n/\delta)}={\mathbb{P}}^ N$$ [for $$\delta >n$$ this implies Hartshorne’s conjecture on linear normality; cf. the author Mat. Sb., Nov. Ser. 116(158), 593-602 (1981; Zbl 0484.14016)].
It should be noted that all varieties X for which the above inequalities turn into equalities have been classified; they are called the Scorza varieties, and for $$\delta =n$$ this notion coincides with the notion of Severi varieties as defined in the author’s paper in Mat. Sbornik 126(168), No.1, 115-132 (1985; see the preceding review).

##### MSC:
 14J40 $$n$$-folds ($$n>4$$) 14N05 Projective techniques in algebraic geometry 14M07 Low codimension problems in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves
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##### References:
 [1] T. Fujita and J. Roberts, ”Varieties with small secant varieties: the extremal case,” Am. J. Math.,103, No. 5, 953-976 (1981). · Zbl 0475.14046 · doi:10.2307/2374254 [2] W. Fulton and R. Lazarsfeld, ”Connectivity and its applications in algebraic geometry,” in: Proc. Midwest Conf. on Algebraic Geometry, Univ. of Illinois, 1980, Lecture Notes in Math., Vol. 862, Springer-Verlag, Berlin?Heidelberg?New York (1981), pp. 26-92. · Zbl 0484.14005 [3] Ph. Griffiths and J. Harris, ”Algebraic geometry and local differential geometry,” Ann. Sci. Ecole Norm. Sup.,12, No. 3, 355-452 (1979). · Zbl 0426.14019 [4] R. Lazarsfeld and A. Van der Ven, ”Topics in the geometry of projective space: Recent work of F. L. Zak,” Birkhäuser Verlag, Basel?Boston?Stuttgart (1984). [5] F. L. Zak, ”Projections of algebraic varieties,” Mat. Sb.,116, No. 4, 593-602 (1981). · Zbl 0484.14016 [6] F. L. Zak, ”Severi varieties,” Mat. Sb.,126, No. 1, 115-132 (1985). · Zbl 0577.14031
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