Algebraic Curves and Riemann Surfaces - R. Miranda
This text has evolved from lecture notes for a one-semester course which I have
taught 5 times in the last 8 years as an introduction to the ideas of algebraic
geometry using the theory of algebraic curves as a foundation.
There are two broad aims for the book: to keep the prerequisites to a bare
minimum while still treating the major theorems seriously; and to begin to con-
vey to the reader some of the language of modern algebraic geometry.
In order to present the material of Algebraic Curves to an initially relatively
unsophisticated audience I have taken the approach that Algebraic Curves are
best encountered for the first time over the complex numbers. Therefore the
book starts out as a primer on Palemann surfaces, with complex charts and mero-
morphic functions taking center stage. In particular, one semester of graduate
complex analysis should be sufficient preparation, and it is not assumed that the
reader has any serious background in either algebraic topology or commutative
algebra. But I try to stress that the main examples (from the point of view
of algebraic geometry) come from projective curves, and slowly but surely the
text evolves to the algebraic category, culminating in an algebraic proof of the
Riemann-Roch theorem. After returning to the analytic side of things for Abel's
theorem, the progression is repeated again when sheaves and cohomology are
discussed: first the analytic, then the algebraic category.
The proof of Riemann-Roch presented here is an adaptation of the adelic
proof, expressed completely in terms of solving a Mittag-Leffier problem. This
is a very concrete approach, and in particular no cohomology or sheaf theory is
used. However, cohomology groups clandestinely appear (as obstruction spaces
to solving Mittag-Leffier problems), motivating their explicit introduction later
on.
- Hardcover: 390 pages
- Publisher: American Mathematical Society (April 1, 1995)
- Language: English
- ISBN-10: 0821802682
- ISBN-13: 978-0821802687
- Product Dimensions: 10.1 x 7 x 1.1 inches
This text has evolved from lecture notes for a one-semester course which I have
taught 5 times in the last 8 years as an introduction to the ideas of algebraic
geometry using the theory of algebraic curves as a foundation.
There are two broad aims for the book: to keep the prerequisites to a bare
minimum while still treating the major theorems seriously; and to begin to con-
vey to the reader some of the language of modern algebraic geometry.
In order to present the material of Algebraic Curves to an initially relatively
unsophisticated audience I have taken the approach that Algebraic Curves are
best encountered for the first time over the complex numbers. Therefore the
book starts out as a primer on Palemann surfaces, with complex charts and mero-
morphic functions taking center stage. In particular, one semester of graduate
complex analysis should be sufficient preparation, and it is not assumed that the
reader has any serious background in either algebraic topology or commutative
algebra. But I try to stress that the main examples (from the point of view
of algebraic geometry) come from projective curves, and slowly but surely the
text evolves to the algebraic category, culminating in an algebraic proof of the
Riemann-Roch theorem. After returning to the analytic side of things for Abel's
theorem, the progression is repeated again when sheaves and cohomology are
discussed: first the analytic, then the algebraic category.
The proof of Riemann-Roch presented here is an adaptation of the adelic
proof, expressed completely in terms of solving a Mittag-Leffier problem. This
is a very concrete approach, and in particular no cohomology or sheaf theory is
used. However, cohomology groups clandestinely appear (as obstruction spaces
to solving Mittag-Leffier problems), motivating their explicit introduction later
on.
