Hijikata / Hironaka / Maruyama Algebraic Geometry and Commutative Algebra

In Honor of Masayoshi Nagata
1. Auflage 2014
ISBN: 978-1-4832-6505-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 406 Seiten, Web PDF

Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata presents a collection of papers on algebraic geometry and commutative algebra in honor of Masayoshi Nagata for his significant contributions to commutative algebra. Topics covered range from Weierstrass models and endomorphism algebras of abelian varieties to the generic Torelli theorem for hypersurfaces in compact irreducible hermitian symmetric spaces. Coarse moduli spaces for curves are also discussed, along with discriminants of curves of genus 2 and arithmetic surfaces. Comprised of 14 chapters, this volume begins by describing a basic fibration as a Weierstrass model, with emphasis on elliptic threefolds with a section. The reader is then introduced to canonical bundles of analytic surfaces of class VII0 with curves; Lifting Problem on ideal-adically complete noetherian rings; and the canonical ring of a curve. Subsequent chapters deal with algebraic surfaces for regular systems of weights; elementary transformations of algebraic vector bundles; the irreducibility of the first differential equation of Painlevé; and F-pure normal rings of dimension two. The book concludes with an assessment of the existence of some curves. This monograph will be a useful resource for practitioners and researchers in algebra and geometry.

Autoren/Hrsg.

Hijikata, Hiroaki

Hironaka, Heisuke

Maruyama, Masaki

Inhaltsverzeichnis

1;Front Cover ;1
2;Algebraic Geometry and Commutative Algebra: In Honor of Masayoshi NAGATA;4
3;Copyright Page;5
4;Table of Contents;6
5;Table of Contents of Volume I;7
6;Chapter 1. On Weierstrass Models;8
6.1;Introduction;8
6.2;Notation;8
6.3;0. Variation of Hodge structures;9
6.4;1. Elementary properties;12
6.5;2. Elliptic fibrations;13
6.6;3. Elliptic threefolds;22
6.7;References;29
6.8;Appendix;30
6.8.1;1. Lemmas on ramifications;31
6.8.2;2. Main Theorem;32
6.8.3;References;34
7;Chapter 2. Canonical Bundles of Analytic Surfaces of Class VII;36
7.1;Introduction;36
7.2;Notation and Convention;37
7.3;1.;37
7.4;2.;39
7.5;3.;43
7.6;4.;44
7.7;5.;46
7.8;6.;47
7.9;References;54
8;Chapter 3. Ideal-adic Completion of Noetherian Rings II;56
8.1;Introduction;56
8.2;Notation;58
8.3;1. Rotthaus' Hilfssatz. (cf. [19], [11]);58
8.4;2. Proof of Theorem A;64
8.5;3. Proof of Theorem B;66
8.6;References;69
9;Chapter 4. Endomorphism Algebras of Abelian Varieties;72
9.1;Introduction;72
9.2;1. Notations;74
9.3;2. Elliptic curves;79
9.4;3. Some general facts. Types I and II;82
9.5;4. Type III;91
9.6;5. Type IV;94
9.7;6. Abelian surfaces;97
9.8;7. The case dim X = g, a prime number g > 2;99
9.9;8. Survey of the results. Some questions;100
9.10;References;102
10;Chapter 5. On the Canonical Ring of a Curve;106
10.1;1. Introduction;106
10.2;2. Clifford Index and Green's Conjecture;107
10.3;3. Stability Properties of E;110
10.4;4. Locally Decomposable Sections of ^dE;116
10.5;References;119
11;Chapter 6. Algebraic Surfaces for Regular Systems of Weights;120
11.1;Abstract;120
11.2;Contents;120
11.3;1. Introduction;120
11.4;2. System of weights, having one negative exponent;124
11.5;3. System of weights, having one negative exponent and some 0 exponents;168
11.6;4. System of weights, whose smallest exponent e is equal to —2;173
11.7;5. Weighted homogeneous singularity of dimension two;188
11.8;Appendix;198
11.9;Appendix A: Rational elliptic surfaces for D4, E6, E7 and E8;199
11.10;References;215
12;Chapter 7. Generic Torelli Theorem for Hypersurfaces in Compact Irreducible Hermitian Symmetric Spaces;218
12.1;0. Introduction;218
12.2;1. Preliminaries;220
12.3;2. Vanishing theorems for the cohomology groups Hq(Y,Opy(k));227
12.4;3. Jacobian rings and the duality theorem;234
12.5;4. Infinitesimal variation of Hodge structure of hypersurfaces;244
12.6;5. Polynomial structure;251
12.7;6. Symmetrizer lemma;252
12.8;7. Moduli space and the period map;255
12.9;8. Generic Torelli theorem;257
12.10;9. Appendix. (Proof of theorem (2.3.1));260
12.11;References;266
13;Chapter 8. A Variety Which Contains a P1-fiber Space as an Ample Divisor;268
13.1;Introduction;268
13.2;Conventions and notations;269
13.3;1. Separably uniruled varieties;270
13.4;2. Uniruled varieties with UR(X) = n and non-zero sections in H0(X,Sm(^nOX) );276
13.5;3. Fiber structure of uniruled varieties;279
13.6;4. Variety whose hyperplane section is a P1-bundle;283
13.7;References;293
14;Chapter 9. How Coarse the Coarse Moduli Spaces for Curves Are!;296
14.1;0. Introduction;296
14.2;1. Hyperelliptic curves with an automorphism of order p;297
14.3;2. Kodaira-Spencer map for a hyperellliptic curve;303
14.4;3. Main results;310
14.5;References;314
15;Chapter 10. Elementary Transformations of Algebraic Vector Bundles II;316
15.1;0. Introduction;316
15.2;1. A construction method of algebraic vector bundles;317
15.3;2. Some properties of E(Z,W1,...,Wr);328
15.4;3. An example (Horrocks-Mumford bundle);341
15.5;References;350
16;Chapter 11. Discriminants of Curves of Genus 2 and Arithmetic Surfaces;352
16.1;0. Introduction;352
16.2;1. A pencil of curves of genus 1;354
16.3;2. Pencils of curves of genus 2;356
16.4;3. Anaytic theory of pencils of curves of genus 2;361
16.5;4. Arithmetic surfaces;365
16.6;References;372
17;Chapter 12. On the Irreducibility of the First Differential Equation of Painlevé;374
17.1;1. Review of the preceding paper;375
17.2;2. Proof of the Theorem;377
17.3;References;391
18;Chapter 13. Study of F-purity in Demension Two;394
18.1;Introduction;394
18.2;0. Preliminaries and notations;395
18.3;1. Gorenstein case;396
18.4;2. Examples of rational singularities which are not of F-pure type;399
18.5;References;402
19;Chapter 14. A Note on the Existence of Some Curves;404
19.1;1. Introduction;404
19.2;2. An inequality;404
19.3;3. Proofs;406
19.4;References;407