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Softcover ISBN:  9781470479435 
eBook: ISBN:  9781470479442 
Product Code:  GSM/250.S.B 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Hardcover ISBN:  9781470476373 
Product Code:  GSM/250 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Softcover ISBN:  9781470479435 
Product Code:  GSM/250.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470479442 
Product Code:  GSM/250.E 
List Price:  $0.00 
MAA Member Price:  $0.00 
AMS Member Price:  $0.00 
Softcover ISBN:  9781470479435 
eBook ISBN:  9781470479442 
Product Code:  GSM/250.S.B 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 

Book DetailsGraduate Studies in MathematicsVolume: 250; 2024; 413 ppMSC: Primary 14; 13
NOTE: For this title, the eBook is available for free as part of a pilot program to promote access to key mathematical texts. We are collecting information about the readers reached through the AMS Bookstore. These data will only be reported and analyzed at the aggregate level in order to measure the success of the pilot.
This textbook provides readers with a working knowledge of the modern theory of complex projective algebraic curves. Also known as compact Riemann surfaces, such curves shaped the development of algebraic geometry itself, making this theory essential background for anyone working in or using this discipline. Examples underpin the presentation throughout, illustrating techniques that range across classical geometric theory, modern commutative algebra, and moduli theory.
The book begins with two chapters covering basic ideas, including maps to projective space, invertible sheaves, and the Riemann–Roch theorem. Subsequent chapters alternate between a detailed study of curves up to genus six and more advanced topics such as Jacobians, Hilbert schemes, moduli spaces of curves, Severi varieties, dualizing sheaves, and linkage of curves in 3space. Three chapters treat the refinements of the Brill–Noether theorem, including applications and a complete proof of the basic result. Two chapters on free resolutions, rational normal scrolls, and canonical curves build context for Green’s conjecture. The book culminates in a study of Hilbert schemes of curves through examples. A historical appendix by Jeremy Gray captures the early development of the theory of algebraic curves. Exercises, illustrations, and open problems accompany the text throughout.
The Practice of Algebraic Curves offers a masterclass in theory that has become essential in areas ranging from algebraic geometry itself to mathematical physics and other applications. Suitable for students and researchers alike, the text bridges the gap from a first course in algebraic geometry to advanced literature and active research.
ReadershipGraduate students considering working in the field of algebraic curves and researchers in a related field whose work has led them to questions about algebraic curves.

Table of Contents

Chapters

Introduction

Linear series and morphisms to projective space

The RiemannRoch theorem

Curves of genus 0

Smooth plane curves and curves of genus 1

Jacobians

Hyperelliptic curves and curves of genus 2 and 3

Fine moduli spaces

Moduli of curves

Curves of genus 4 and 5

Hyperplane sections of a curve

Monodromy of hyperplane sections

BrillNoether theory and applications to genus 6

Inflection points

Proof of the BrillNoether theorem

Using a singular plane model

Linkage and the canonical sheave of a singular curves

Scrolls and the curves they contain

Free resolutions and canonical curves

Hilbert schemes

Appendix A. A historical essay on some topics in algebraic geometry (by Jeremy Gray)

Hints to marked exercises


Additional Material

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NOTE: For this title, the eBook is available for free as part of a pilot program to promote access to key mathematical texts. We are collecting information about the readers reached through the AMS Bookstore. These data will only be reported and analyzed at the aggregate level in order to measure the success of the pilot.
This textbook provides readers with a working knowledge of the modern theory of complex projective algebraic curves. Also known as compact Riemann surfaces, such curves shaped the development of algebraic geometry itself, making this theory essential background for anyone working in or using this discipline. Examples underpin the presentation throughout, illustrating techniques that range across classical geometric theory, modern commutative algebra, and moduli theory.
The book begins with two chapters covering basic ideas, including maps to projective space, invertible sheaves, and the Riemann–Roch theorem. Subsequent chapters alternate between a detailed study of curves up to genus six and more advanced topics such as Jacobians, Hilbert schemes, moduli spaces of curves, Severi varieties, dualizing sheaves, and linkage of curves in 3space. Three chapters treat the refinements of the Brill–Noether theorem, including applications and a complete proof of the basic result. Two chapters on free resolutions, rational normal scrolls, and canonical curves build context for Green’s conjecture. The book culminates in a study of Hilbert schemes of curves through examples. A historical appendix by Jeremy Gray captures the early development of the theory of algebraic curves. Exercises, illustrations, and open problems accompany the text throughout.
The Practice of Algebraic Curves offers a masterclass in theory that has become essential in areas ranging from algebraic geometry itself to mathematical physics and other applications. Suitable for students and researchers alike, the text bridges the gap from a first course in algebraic geometry to advanced literature and active research.
Graduate students considering working in the field of algebraic curves and researchers in a related field whose work has led them to questions about algebraic curves.

Chapters

Introduction

Linear series and morphisms to projective space

The RiemannRoch theorem

Curves of genus 0

Smooth plane curves and curves of genus 1

Jacobians

Hyperelliptic curves and curves of genus 2 and 3

Fine moduli spaces

Moduli of curves

Curves of genus 4 and 5

Hyperplane sections of a curve

Monodromy of hyperplane sections

BrillNoether theory and applications to genus 6

Inflection points

Proof of the BrillNoether theorem

Using a singular plane model

Linkage and the canonical sheave of a singular curves

Scrolls and the curves they contain

Free resolutions and canonical curves

Hilbert schemes

Appendix A. A historical essay on some topics in algebraic geometry (by Jeremy Gray)

Hints to marked exercises