Hijikata / Hironaka / Maruyama Algebraic Geometry and Commutative Algebra

In Honor of Masayoshi Nagata
1. Auflage 2014
ISBN: 978-1-4832-6518-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 416 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 power series rings and rings of invariants of finite linear groups to the convolution algebra of distributions on totally disconnected locally compact groups. The discussion begins with a description of several formulas for enumerating certain types of objects, which may be tabular arrangements of integers called Young tableaux or some types of monomials. The next chapter explains how to establish these enumerative formulas, with emphasis on the role played by transformations of determinantal polynomials and recurrence relations satisfied by them. The book then turns to several applications of the enumerative formulas and universal identity, including including enumerative proofs of the straightening law of Doubilet-Rota-Stein and computations of Hilbert functions of polynomial ideals of certain determinantal loci. Invariant differentials and quaternion extensions are also examined, along with the moduli of Todorov surfaces and the classification problem of embedded lines in characteristic p. 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;Foreword;8
5;Table of Contents of Volume II;13
6;Determinantal Loci and Enumerative Combinatorics of Young Tableaux;14
6.1;1. Introduction;14
6.2;First Chapter. YOUNG TABLEAUX AND DETERMINANTAL POLYNOMIALS IN BINOMIAL COEFFICIENTS;15
6.2.1;2. Tableaux and monomials;15
6.2.2;3. Determinantal polynomials of any width;18
6.2.3;4. Determinantal polynomials of width two;20
6.3;Second Chapter.
ENUMERATION OF YOUNG TABLEAUX;23
6.3.1;5. Counting tableaux of any width;23
6.3.2;6. Bitableaux;24
6.3.3;7. Counting bitableaux;24
6.3.4;8. Counting monomials;24
6.3.5;9. Bitableaux and monomials;25
6.4;Third Chapter.
UNIVERSAL DETERMINANTAL IDENTITY;26
6.4.1;10. Preamble;26
6.4.2;11. The mixed size case;26
6.4.3;12. The cardinality condition;28
6.4.4;13. The maximal size case;29
6.4.5;14. The basic case;29
6.4.6;15. Laplace development;29
6.4.7;16. The full depth case;29
6.4.8;17. Deduction of the full depth case;30
6.4.9;18. The straightening law;31
6.4.10;19. Problem;31
6.5;Fourth Chapter.
APPLICATIONS TO IDEAL THEORY;31
6.5.1;20. Determinantal loci;31
6.5.2;21. Vector spaces and homogeneous rings;35
6.5.3;22. Standard basis;36
6.5.4;23. Second fundamental theorem of invariant theory;37
6.5.5;24. Generalized second fundamental theorem of invariant theory;37
6.6;References;39
7;A Conjecture of Sharp —The Case of Local Rings with dim non CM = 1 or dim = 5;40
7.1;1. Introduction;40
7.2;2. Sharp's Conjecture;40
7.3;3. Proofs of Theorem 1.1 and Theorem 1.2;42
7.4;References;46
8;A Structure Theorem for Power Series Rings;48
8.1;1. We suppose that there is given a commutative diagram;50
8.2;2. We may replace B by C = R[X,Y]/(f1,...„fm);50
8.3;3.;51
8.4;4. Proof of the Theorem;53
8.5;5. Corollary;55
8.6;References;56
9;On Rational Plane Sextics with Six Tritangents Wolf BARTH* and Ross MOORE;58
9.1;0. Introduction;58
9.2;1. Some Polynomials;59
9.3;2. The sextic space curve S;60
9.4;3. The projected curves Sx ;63
9.5;4. The double plane X;64
9.6;5. The double plane Y;67
9.7;6. Moduli;68
9.8;7. Explanations;70
9.9;References;71
10;On Rings of Invariants of Finite Linear Groups;72
10.1;1. Fundamental groups;72
10.2;2. Proof of Theorem A;74
10.3;3. Additional results;75
10.4;References;77
11;Invariant Differentials;78
11.1;§1. Introduction;78
11.2;2. Use of the étale slice theorem;79
11.3;3. The ñnite group case;80
11.4;References;84
12;Classification of Polarized Manifoldsof Sectional Genus Two;86
12.1;Introduction;86
12.2;Notation, Convention and Terminology;87
12.3;1. Classification, first step;87
12.4;2. The case K ~ (3 – n)L;90
12.5;3. The case of a hyperquadric fíbration over a curve;96
12.6;4. Polarized surfaces of sectional genus two;104
12.7;Appendix;108
12.8;References;109
13;Affine Surfaces with . = 1;112
13.1;Introduction;112
13.2;1. Surfaces with K = –8;113
13.3;2. The case K{S) = 0;115
13.4;3. The case K{S) = 1;118
13.5;4. Examples K{S) = 2;133
13.6;References;137
14;On the Convolution Algebra of Distributionson Totally Disconnected Locally Compact Groups;138
14.1;0. Introduction;138
14.2;1. Finite w-distribution;139
14.3;2. Action of homeomorphisms and multiplication by functions;140
14.4;3. Generators of S(X, w; V);142
14.5;4. Action of . on vector valued functions;143
14.6;5. Tensor product of distributions;144
14.7;6. Convolution;145
14.8;7. Representation of G;146
14.9;8. Regular representation;146
14.10;9. Projection operator;147
14.11;10. D-modules and S.-modules;148
14.12;11. D-modules and e-modules;149
14.13;12. Proof of Theorem 1;151
14.14;13. Proof of Theorem 2;151
14.15;References;153
15;The Local Cohomology Groups of an Affine Semigroup Ring;154
15.1;Introduction;154
15.2;1. Affine semigroup rings and the associated cones;154
15.3;2. Complexes associated to an affine semigroup ring;158
15.4;3. The dualizing complex and the local cohomology groups;160
15.5;4. Serre's condition (S2);163
15.6;References;165
16;Quaternion Extensions;168
16.1;Definitions, Notations and Some Necessary Facts;168
16.2;Introduction.;169
16.3;I. Quaternion extensions and quadratic forms;170
16.4;II. Fields that admit quaternion extensions;180
16.5;III. Automatic realizations;188
16.6;IV. Polynomials with Galois group Qs;192
16.7;Acknowledgement.;194
16.8;References;194
17;On the Discriminants of the Intersection Form on Néron-Severi Groups;196
17.1;0. Introduction;196
17.2;1. Preliminaries;197
17.3;2. Bilinear forms;199
17.4;3. The Discriminant of the intersection form;202
17.5;4. Examples;205
17.6;5. A K3 surface;208
17.7;References;213
18;On Complete Ideals in Regular Local Rings;216
18.1;Introduction;216
18.2;1. Point bases and completions of ideals in regular local rings;217
18.3;2. Simple complete ideals corresponding to infinitely near points;229
18.4;3. The length of a complete ideal (dimension 2);235
18.5;4. Unique factorization for complete ideals (dimension 2);240
18.6;References;242
19;On a Compactification of a Moduli Space of Stable Vector Bundles on a Rational Surface;246
19.1;Introduction.;246
19.2;1. Some remarks on semi-stable sheaves;248
19.3;2. Semi-stable sheaves on a rational surface;250
19.4;3. Semi-stability of the universal extension;253
19.5;4. Image of . ( G, C1 , C2);258
19.6;5. Image of .(r,0,C2);261
19.7;6. Good polarizations and the case of rank 2;270
19.8;References;272
20;On the Dimension of Formal Fibres of a Local Ring;274
20.1;Introduction.;274
20.2;1. Formal fibres;274
20.3;2. Some cases where a{A) is smaller than dim A – 1;276
20.4;References;279
21;On the Classification Problem of Embedded Lines in Characteristic .;280
21.1;1. Introduction;280
21.2;2. Expansions and Their Calculus;282
21.3;3. The Defining Equations;284
21.4;4. ai = 0;288
21.5;5. Other Coiijectiires;291
21.6;References;292
22;A Cancellation Theorem for Projective Modules over Finitely Generated Rings;294
22.1;1. Introduction;294
22.2;2. Cancellation;294
22.3;3. Projective stable ranges;297
22.4;References;299
23;Semi-ampleness of the Numerically Effective Part of Zariski Decomposition II;302
23.1;0. Introduction;302
23.2;1. Preliminary;302
23.3;2. Zariski decomposition;306
23.4;3. Canonical rings;319
23.5;References;323
24;On the Moduli of Todorov Surfaces;326
24.1;1. Equidistant binary linear codes;328
24.2;2. K3 surfaces with ordinary double points;332
24.3;3 . Double covers of surfaces with ordinary double points;338
24.4;4. Involutions on canonical surfaces;342
24.5;5. Todorov surfaces;344
24.6;6. Embeddings of Todorov lattices;349
24.7;7. The moduli of Todorov surfaces;357
24.8;8. Concluding remarks;363
24.9;Appendix;365
24.10;References;366
25;Curves, K3 Surfaces and Fano 3-folds of Genus = 10;370
25.1;1. Preliminary;373
25.2;2. Proof of Theorem 0.2 in the case g = 10;377
25.3;3. Generic K3 surfaces of genus 7,8, and 9;380
25.4;4. Generic K3 surface of genus 6;384
25.5;5. Fano 3-folds of genus 10;387
25.6;6. Curves of genus = 9;388
25.7;References;389
26;Threefolds Homeomorphic toa Hyperquadric in P4;392
26.1;0. Introduction;392
26.2;1. Hyperquadrics in P4;394
26.3;2. Lemmas;395
26.4;3. A complete intersection L = DnD;396
26.5;4. Proof of (3.2);400
26.6;5. Proof of (3.3);401
26.7;6. Proof of (3.4);403
26.8;7. Proof of (3.5);407
26.9;8. Proof of (3.6);408
26.10;9. Proof of (0.1);414
26.11;Appendix;415
26.12;References;417