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.
