Commutative algebra and its interactions to algebraic geometry. VIASM 2013-2014

Preface......Page 6
Contents......Page 8
Contributors......Page 9
 1.1 Introduction......Page 10
 1.2 Filtered Algebras......Page 17
 1.3 Associated Graded Rings......Page 21
 1.4 Derivations......Page 27
 1.5 Weyl Algebras......Page 30
 1.6 Arithmetic in Weyl Algebras......Page 34
 1.7 The Standard Basis......Page 43
 1.8 Weighted Degrees and Filtrations......Page 50
 1.9 Weighted Associated Graded Rings......Page 56
 1.10 Filtered Modules......Page 63
 1.11 D-Modules......Page 69
 1.12 Gröbner Bases......Page 81
 1.13 Weighted Orderings......Page 98
 1.14 Standard Degree and Hilbert Polynomials......Page 113
 References......Page 124
 2.1 Introduction......Page 127
 2.2 Injective Modules: Matlis\' Duality......Page 129
 2.3 Macaulay\'s Correspondence......Page 138
 2.4 Gorenstein, Level and Compressed Algebras......Page 143
 2.5 Classification of Artin Rings......Page 148
 2.6 Computation of Betti Numbers......Page 158
 2.7 Examples......Page 166
 References......Page 170
 3.1 Introduction......Page 172
 3.2 Preliminaries......Page 176
 3.3 The Representation Type of a del Pezzo Surface......Page 180
  3.3.1 Mustaţă\'s Conjecture for a Set of General Points on a del Pezzo Surface......Page 182
  3.3.2 Ulrich Bundles on del Pezzo Surfaces......Page 195
 3.4 The Representation Type of a Segre Variety......Page 201
  3.4.1 Representation Type of Σn1,... ,ns, 2≤n1, ... ,ns......Page 204
  3.4.2 Representation Type of Σn1,n2...,ns, 1=n1≤n2, ... , ns......Page 209
 3.5 Does the Representation Type of a Projective Variety Depends on the Polarization?......Page 215
 3.6 Open Problems......Page 220
 References......Page 221
 4.1 Introduction......Page 224
 4.2 Definition of Toric Varieties by Parametrization, Semigroups or Lattices......Page 227
 4.3 Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections......Page 229
  4.3.1 Lattice of Relations of Simplicial Toric Varieties......Page 230
  4.3.2 Simplicial Toric Varieties in Characteristic p>0......Page 234
  4.3.3 Almost Set-Theoretic Complete Intersections......Page 236
 4.4 Equations in Codimension 2......Page 237
  4.4.1 The Lattice Associated in Codimension Two......Page 238
  4.4.2 Effective Computation of the Fan Associated to the Universal Grobner Basis of IA......Page 241
  4.5.1 Almost-Complete Intersections: The General Case......Page 244
  4.5.2 Almost-Complete Intersections, The Codimension Two Case......Page 250
  4.6.1 Tricks on Toric Varieties......Page 252
  4.6.2 Toric Curves in P3......Page 254
  4.6.3 Toric Curves in Pn......Page 258
 References......Page 262
LECTURE NOTES IN MATHEMATICS......Page 264