Ebook: Projective Geometry and Formal Geometry

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 Ysubset 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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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 Ysubset 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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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 Ysubset 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.

Content:
Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Extensions of Projective Varieties....Pages 3-13
Proof of Theorem 1.3....Pages 15-21
Counterexamples and Further Consequences....Pages 23-30
The Zak Map of a Curve. Gaussian Maps....Pages 31-38
Quasi-homogeneous Singularities and Projective Geometry....Pages 39-48
A Characterization of Linear Subspaces....Pages 49-54
Cohomological Dimension and Connectedness Theorems....Pages 55-67
A Problem of Complete Intersection....Pages 69-78
Front Matter....Pages 79-79
Basic Definitions and Results....Pages 81-109
Lefschetz Theory and Meromorphic Functions....Pages 111-121
Connectedness and Formal Functions....Pages 123-144
Further Results on Formal Functions....Pages 145-156
Formal Functions on Homogeneous Spaces....Pages 157-174
Quasi-lines on Projective Manifolds....Pages 175-194
Back Matter....Pages 195-214

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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 Ysubset 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.

Content:
Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Extensions of Projective Varieties....Pages 3-13
Proof of Theorem 1.3....Pages 15-21
Counterexamples and Further Consequences....Pages 23-30
The Zak Map of a Curve. Gaussian Maps....Pages 31-38
Quasi-homogeneous Singularities and Projective Geometry....Pages 39-48
A Characterization of Linear Subspaces....Pages 49-54
Cohomological Dimension and Connectedness Theorems....Pages 55-67
A Problem of Complete Intersection....Pages 69-78
Front Matter....Pages 79-79
Basic Definitions and Results....Pages 81-109
Lefschetz Theory and Meromorphic Functions....Pages 111-121
Connectedness and Formal Functions....Pages 123-144
Further Results on Formal Functions....Pages 145-156
Formal Functions on Homogeneous Spaces....Pages 157-174
Quasi-lines on Projective Manifolds....Pages 175-194
Back Matter....Pages 195-214
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