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Reembeddings of special border basis schemes

  • Martin Kreuzer und Lorenzo Robbiano
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Commutative Algebra
Ein Kapitel aus dem Buch Commutative Algebra

Abstract

Border basis schemes are open subschemes of the Hilbert scheme of μ points in an affine space A n . They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space A μ ν . Here we bring together several techniques for reembedding affine schemes into lower-dimensional spaces which we developed in the last years. We study their efficacy for some special types of border basis schemes such as MaxDeg border basis schemes, L-shape and simplicial border basis schemes, as well as planar border basis schemes. Particular care is taken to make these reembeddings efficiently computable and to check when we actually get an isomorphism with A n μ , i. e., when the border basis scheme is an affine cell.

Abstract

Border basis schemes are open subschemes of the Hilbert scheme of μ points in an affine space A n . They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space A μ ν . Here we bring together several techniques for reembedding affine schemes into lower-dimensional spaces which we developed in the last years. We study their efficacy for some special types of border basis schemes such as MaxDeg border basis schemes, L-shape and simplicial border basis schemes, as well as planar border basis schemes. Particular care is taken to make these reembeddings efficiently computable and to check when we actually get an isomorphism with A n μ , i. e., when the border basis scheme is an affine cell.

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Wolmer Vasconcelos, a tribute IX
  5. Vasconcelos Bibliography (1964–2024) XVII
  6. The early years: flatness, projectivity, and R-sequences 1
  7. Joint work with Wolmer Vasconcelos. The early years 41
  8. Homological properties of the module of differentials 71
  9. Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
  10. Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
  11. Degrees: Vasconcelos contributions 203
  12. The books of Wolmer V. Vasconcelos 241
  13. Finding nilpotents with Wolmer Vasconcelos 261
  14. Rees algebras over residual intersections 279
  15. Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
  16. Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
  17. Reembeddings of special border basis schemes 353
  18. Z r -graded rings and their canonical modules 389
  19. Rings with q-torsionfree canonical modules 405
https://doi.org/10.1515/9783110999365-012

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Wolmer Vasconcelos, a tribute IX
  5. Vasconcelos Bibliography (1964–2024) XVII
  6. The early years: flatness, projectivity, and R-sequences 1
  7. Joint work with Wolmer Vasconcelos. The early years 41
  8. Homological properties of the module of differentials 71
  9. Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
  10. Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
  11. Degrees: Vasconcelos contributions 203
  12. The books of Wolmer V. Vasconcelos 241
  13. Finding nilpotents with Wolmer Vasconcelos 261
  14. Rees algebras over residual intersections 279
  15. Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
  16. Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
  17. Reembeddings of special border basis schemes 353
  18. Z r -graded rings and their canonical modules 389
  19. Rings with q-torsionfree canonical modules 405
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Heruntergeladen am 10.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/9783110999365-012/html
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