Ebook: Monomialization of Morphisms from 3-folds to Surfaces

A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally be represented in e'tale neighborhoods by a pure monomial mappings. The book gives proof that a dominant morphism from a nonsingular 3-fold X to a surface S can be monomialized by performing sequences of blowups of nonsingular subvarieties of X and S.
The construction is very explicit and uses techniques from resolution of singularities. A research monograph in algebraic geometry, it addresses researchers and graduate students.

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A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally be represented in e'tale neighborhoods by a pure monomial mappings. The book gives proof that a dominant morphism from a nonsingular 3-fold X to a surface S can be monomialized by performing sequences of blowups of nonsingular subvarieties of X and S.
The construction is very explicit and uses techniques from resolution of singularities. A research monograph in algebraic geometry, it addresses researchers and graduate students.

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A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally be represented in e'tale neighborhoods by a pure monomial mappings. The book gives proof that a dominant morphism from a nonsingular 3-fold X to a surface S can be monomialized by performing sequences of blowups of nonsingular subvarieties of X and S.
The construction is very explicit and uses techniques from resolution of singularities. A research monograph in algebraic geometry, it addresses researchers and graduate students.
Content:
1. Introduction....Pages 1-8
2. Local Monomialization....Pages 9-10
3. Monomialization of Morphisms in Low Dimensions....Pages 11-13
4. An Overview of the Proof of Monomialization of Morphisms From 3 Folds to Surfaces....Pages 14-18
5. Notations....Pages 19-19
6. The Invariant $nu$ ....Pages 20-55
7. The Invariant $nu$ Under Quadratic Transforms....Pages 56-76
8. Permissible Monoidal Transforms Centered at Curves....Pages 77-92
9. Power Series in 2 Variables....Pages 93-108
10. $bf {A_r(X)}$ ....Pages 109-109
11. Reduction of $nu$ in a Special Case....Pages 110-130
12. Reduction of $nu$ in a Second Special Case....Pages 131-149
13. Resolution 1....Pages 150-162
14. Resolution 2....Pages 163-175
15. Resolution 3....Pages 176-184
16. Resolution 4....Pages 185-187
17. Proof of the Main Theorem....Pages 188-188
18. Monomialization....Pages 189-223
19. Toroidalization....Pages 224-231
20. Glossary of Notations and Definitions....Pages 232-233

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A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally be represented in e'tale neighborhoods by a pure monomial mappings. The book gives proof that a dominant morphism from a nonsingular 3-fold X to a surface S can be monomialized by performing sequences of blowups of nonsingular subvarieties of X and S.
The construction is very explicit and uses techniques from resolution of singularities. A research monograph in algebraic geometry, it addresses researchers and graduate students.
Content:
1. Introduction....Pages 1-8
2. Local Monomialization....Pages 9-10
3. Monomialization of Morphisms in Low Dimensions....Pages 11-13
4. An Overview of the Proof of Monomialization of Morphisms From 3 Folds to Surfaces....Pages 14-18
5. Notations....Pages 19-19
6. The Invariant $nu$ ....Pages 20-55
7. The Invariant $nu$ Under Quadratic Transforms....Pages 56-76
8. Permissible Monoidal Transforms Centered at Curves....Pages 77-92
9. Power Series in 2 Variables....Pages 93-108
10. $bf {A_r(X)}$ ....Pages 109-109
11. Reduction of $nu$ in a Special Case....Pages 110-130
12. Reduction of $nu$ in a Second Special Case....Pages 131-149
13. Resolution 1....Pages 150-162
14. Resolution 2....Pages 163-175
15. Resolution 3....Pages 176-184
16. Resolution 4....Pages 185-187
17. Proof of the Main Theorem....Pages 188-188
18. Monomialization....Pages 189-223
19. Toroidalization....Pages 224-231
20. Glossary of Notations and Definitions....Pages 232-233
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