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Koszul Cohomology and Algebraic Geometry
 
Marian Aprodu Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, Bucharest, Romania
Jan Nagel Université de Bourgogne, Dijon, France
Koszul Cohomology and Algebraic Geometry
Softcover ISBN:  978-0-8218-4964-4
Product Code:  ULECT/52
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-1647-8
Product Code:  ULECT/52.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-4964-4
eBook: ISBN:  978-1-4704-1647-8
Product Code:  ULECT/52.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
Koszul Cohomology and Algebraic Geometry
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Koszul Cohomology and Algebraic Geometry
Marian Aprodu Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, Bucharest, Romania
Jan Nagel Université de Bourgogne, Dijon, France
Softcover ISBN:  978-0-8218-4964-4
Product Code:  ULECT/52
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-1647-8
Product Code:  ULECT/52.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-4964-4
eBook ISBN:  978-1-4704-1647-8
Product Code:  ULECT/52.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
  • Book Details
     
     
    University Lecture Series
    Volume: 522010; 125 pp
    MSC: Primary 14; 13; 16;

    The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well.

    This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

    Readership

    Graduate students and research mathematicians interested in algebraic geometry and Hodge theory.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. Basic definitions
    • Chapter 2. Basic results
    • Chapter 3. Syzygy schemes
    • Chapter 4. The conjectures of Green and Green-Lazarsfeld
    • Chapter 5. Koszul cohomology and the Hilbert scheme
    • Chapter 6. Koszul cohomology of a $K3$ surface
    • Chapter 7. Specific versions of the syzygy conjectures
    • Chapter 8. Applications
  • Reviews
     
     
    • It can best be viewed as a state-of-the-art review of a number of recent results and an encouragement for further research in this area.

      LMS Newsletter
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 522010; 125 pp
MSC: Primary 14; 13; 16;

The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well.

This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

Readership

Graduate students and research mathematicians interested in algebraic geometry and Hodge theory.

  • Chapters
  • Chapter 1. Basic definitions
  • Chapter 2. Basic results
  • Chapter 3. Syzygy schemes
  • Chapter 4. The conjectures of Green and Green-Lazarsfeld
  • Chapter 5. Koszul cohomology and the Hilbert scheme
  • Chapter 6. Koszul cohomology of a $K3$ surface
  • Chapter 7. Specific versions of the syzygy conjectures
  • Chapter 8. Applications
  • It can best be viewed as a state-of-the-art review of a number of recent results and an encouragement for further research in this area.

    LMS Newsletter
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.