Meromorphic Functions and Projective Curves

1. Front Matter

   Pages i-vii

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2. Foundational Material

   • Kichoon Yang

   Pages 1-35

3. Analytic and Algebraic Families

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   Pages 37-53

4. Meromorphic Functions

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   Pages 55-97

5. Brill-Noether Theory

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   Pages 99-123

6. Projective Differential Geometry

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   Pages 125-167

7. Metric Geometry of Curves

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   Pages 169-196

8. Back Matter

   Pages 197-207

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This book contains an exposition of the theory of meromorphic functions and linear series on a compact Riemann surface. Thus the main subject matter consists of holomorphic maps from a compact Riemann surface to complex projective space. Our emphasis is on families of meromorphic functions and holomorphic curves. Our approach is more geometric than algebraic along the lines of [Griffiths-Harrisl]. AIso, we have relied on the books [Namba] and [Arbarello-Cornalba-Griffiths-Harris] to agreat exten- nearly every result in Chapters 1 through 4 can be found in the union of these two books. Our primary motivation was to understand the totality of meromorphic functions on an algebraic curve. Though this is a classical subject and much is known about meromorphic functions, we felt that an accessible exposition was lacking in the current literature. Thus our book can be thought of as a modest effort to expose parts of the known theory of meromorphic functions and holomorphic curves with a geometric bent. We have tried to make the book self-contained and concise which meant that several major proofs not essential to further development of the theory had to be omitted. The book is targeted at the non-expert who wishes to leam enough about meromorphic functions and holomorphic curves so that helshe will be able to apply the results in hislher own research. For example, a differential geometer working in minimal surface theory may want to tind out more about the distribution pattern of poles and zeros of a meromorphic function.

• Divisor
• Grad
• Meromorphic function
• algebraic curve
• differential geometry

• Department of Mathematics, University of Texas — Pan American, Edinburg, USA

  Kichoon Yang

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