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**Extra resources for C-star-algebras. Hilbert Spaces**

**Sample text**

By Ox(d) we shall denote the restriction of the line bundle Oe(v,)(d) to X, as well as the corresponding sheaf of sections. Such sections can be described in a similar way to the above description for the whole P(V*) by considering regular homogeneous functions on :r -1 (U) where U C X is Zariski open. 7. A projective variety X C P(V*) is linearly normal if and only 4. Further examples and properties of duality 35 if it is projectively isomorphic to the image of F z: for some very ample invertible sheaf s on X.

Let a > 2 be a natural number. Consider the hypersurface X C pm with the affine equation X a1 " ~ - ' ' ' ' ~ - X a m -- 1 (or the homogeneous equation x a1 + . . + X ma = X~)). Introduce an affine chart in p m , consisting of hyperplanes with affine equations of the form )-"~4%1pixi 1. So Pl . . . Pm are coordinates in this chart. 3 shows that the hypersurface dual to X can be defined, in coordinates - - Pl . . . _q_ p~'-~ 4 - . . 4- pm-~ = 1. 7) can be replaced by a polynomial equation of degree a (a - 1)m-1.

The embedding of a linearly normal variety can be described intrinsically, in terms of the variety itself and a certain invertible sheaf on it. Let us recall this correspondence between invertible sheaves and projective embeddings [GH] [Hart]. By an invertible sheaf on an algebraic variety, we mean the sheaf of sections of some algebraic line bundle. We shall not distinguish notationally a line bundle from the corresponding invertible sheaf. Let X be a projective variety and let s be an invertible sheaf on X.