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§5 . Theorem of Hironaka and Matsumura«

In this section we give a theorem of Hironaka and Matsumura which characterizes those subvarieties of the projective space P, that are G3. Also we give a theorem of Hartshorne characterizing those sub- varieties of P whose complements have cohomological dimension < n-1. Finally, we give some new examples of non-algebraizable formal schemes.

Theorem ?,1. (Hironaka-Matsumura [l]. Theorem 3.3, p. 69). bet

Y с p = i> be a closed subscheme. Then Y is G3 in 3P <t—^ y is

connected and of dimension > 1.

Proof . Suppose Y is G3 in JP. Then by Remark 2 before Proposition 1.1, Y is connected. It also follows that Y has dimension > 1.

Conversely , suppose Y is connected and of dimension > 1. Then we will show that Y is G3 in 3P.

First , let Y be an irreducible curve in Y. Then we have

morphisms of formal schemes 3Py ---> P/ ---> IP (where X. denotes

о ^

the formal completion of X along the closed subscheme Y), and hence we have inclusion of fields

k ( 3P ) с к(з^у) с k(P/y ) •

Thus it will be sufficient to show that к(ж ) = k(i>). In other

о words, we may assume that Y is an irreducible curve.

Choose a linear subspace L " с ЗР of dimension n-2 which does

not meet Y. Let V = 3P - L, and let тг: v --->1P be the projection