The Canonical Ring of a Stacky Curve.

Format:

Book

Author/Creator:

Voight, John.

Contributor:

Zureick-Brown, David.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.277

Language:

English

Subjects (All):

Curves, Algebraic.

Physical Description:

1 online resource (156 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Grobner basis. We work in a general algebro-geometric context and treat log canonical and spin canonical rings as well. As an application, we give an explicit presentation for graded rings of modular forms arising from finite-area quotients of the upper half-plane by Fuchsian groups"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Motivation: Petri's theorem

1.2. Orbifold canonical rings

1.3. Rings of modular forms

1.4. Main result

1.5. Extensions and discussion

1.6. Previous work on canonical rings of fractional divisors

1.7. Computational applications

1.8. Generalizations

1.9. Organization and description of proof

1.10. Acknowledgements

Chapter 2. Canonical rings of curves

2.1. Setup

2.2. Terminology

2.3. Low genus

2.4. Basepoint-free pencil trick

2.5. Pointed gin: High genus and nonhyperelliptic

2.6. Gin and pointed gin: Rational normal curve

2.7. Pointed gin: Hyperelliptic

2.8. Gin: Nonhyperelliptic and hyperelliptic

2.9. Summary

Chapter 3. A generalized Max Noether's theorem for curves

3.1. Max Noether's theorem in genus at most 1

3.2. Generalized Max Noether's theorem (GMNT)

3.3. Failure of surjectivity

3.4. GMNT: Nonhyperelliptic curves

3.5. GMNT: Hyperelliptic curves

Chapter 4. Canonical rings of classical log curves

4.1. Main result: Classical log curves

4.2. Log curves: Genus 0

4.3. Log curves: Genus 1

4.4. Log degree 1: Hyperelliptic

4.5. Log degree 1: Nonhyperelliptic

4.6. Exceptional log cases

4.7. Log degree 2

4.8. General log degree

4.9. Summary

Chapter 5. Stacky curves

5.1. Stacky points

5.2. Definition of stacky curves

5.3. Coarse space

5.4. Divisors and line bundles on a stacky curve

5.5. Differentials on a stacky curve

5.6. Canonical ring of a (log) stacky curve

5.7. Examples of canonical rings of log stacky curves

Chapter 6. Rings of modular forms

6.1. Orbifolds and stacky Riemann existence

6.2. Modular forms

Chapter 7. Canonical rings of log stacky curves: genus zero

7.1. Toric presentation

7.2. Effective degrees

7.3. Simplification.

Chapter 8. Inductive presentation of the canonical ring

8.1. The block term order

8.2. Block term order: Examples

8.3. Inductive theorem: large degree canonical divisor

8.4. Main theorem

8.5. Inductive theorems: Genus zero, 2-saturated

8.6. Inductive theorem: By order of stacky point

8.7. Poincaré generating polynomials

Chapter 9. Log stacky base cases in genus 0

9.1. Beginning with small signatures

9.2. Canonical rings for small signatures

9.3. Conclusion

Chapter 10. Spin canonical rings

10.1. Classical case

10.2. Modular forms

10.3. Genus zero

10.4. Higher genus

Chapter 11. Relative canonical algebras

11.1. Classical case

11.2. Relative stacky curves

11.3. Modular forms and application to Rustom's conjecture

Appendix: Tables of canonical rings

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470470944

1470470942

OCLC:

1343249233