Projective Geometry and Formal Geometry

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Springer Science & Business Media, Oct 25, 2004 - Mathematics - 214 pages
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The aim of this monograph is to introduce the reader to modern methods of projective geometry involving certain techniques of formal geometry. Some of these methods are illustrated in the first part through the proofs of a number of results of a rather classical flavor, involving in a crucial way the first infinitesimal neighbourhood of a given subvariety in an ambient variety. Motivated by the first part, in the second formal functions on the formal completion X/Y of X along a closed subvariety Y are studied, particularly the extension problem of formal functions to rational functions.
The formal scheme X/Y, introduced to algebraic geometry by Zariski and Grothendieck in the 1950s, is an analogue of the concept of a tubular neighbourhood of a submanifold of a complex manifold. It is very well suited to study the given embedding Y\subset X. The deep relationship of formal geometry with the most important connectivity theorems in algebraic geometry, or with complex geometry, is also studied. Some of the formal methods are illustrated and applied to homogeneous spaces.
The book contains a lot of results obtained over the last thirty years, many of which never appeared in a monograph or textbook. It addresses to algebraic geometers as well as to those interested in using methods of algebraic geometry.


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Section 1
Section 2
Section 3
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Section 6
Section 7
Section 8
Section 9
Section 10
Section 11
Section 12

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Page 209 - D'. So far we didn't use the hypothesis that Y generates X. This means that G = Gy. Therefore D' • G C D', which is obviously impossible because D
Page 212 - Chapter IV, 18) there is a unique structure of abelian variety on A' such that g becomes an isogeny of abelian varieties. Then by the universal property of Alb(y), there is a unique morphism h: A —> A
Page 212 - G1 in X. Moreover, Y is G3 in X if and only if Y generates X and the canonical morphism f has connected fibers. Proof. Part (i) is trivial, as well as the fact that i is a closed embedding (because / o i coincides with the inclusion Y C X).
Page 209 - Y gg~ l D. Hence the cycles Y • D — 0 and Y -g~ [ D > 0 are numerically equivalent on Y, but this is impossible since Y is a projective variety. This proves claim 1. Claim 2. For every g, h e G Y . P one has D'h~ l g C D'. To prove claim 2, let g, h 6 Gy, p and <J € D' be arbitrary elements. Then 6 € dh~ l G Y , p n D' (because /ie Gy, p ), and so, by claim 1 we get Sh^ 1 G Y , P C D'.

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