Algebraic Geometry - D. Bump
In 1989-1990 I taught a course in Algebraic Geometry at Stanford Univer-
sity, writing up lecture notes. These were revised for publication in 1998. In
1989-90 I covered the materiM in Chapters 1-14 in two quarters, and con-
tinued with a quarter on cohomology of coherent sheaves, lecturing out of
Hartshorne's book.
The aim is to make this a text that can be used in a one year at the graduate
level. I have tried to give complete proofs assuming a background in algebra at
the level one expects from a first or second year graduate student. The point
of view here is that of Serre [23] or Chapter I of Mumford [21]--a variety
is a ringed space locally isomorphic to an affine variety over a field, which is
algebraically closed except in Chapter 14. Although I do not treat schemes I
trust the reader will not find the transition too difficult.
The first eight sections contain material applicable to varieties of every
dimension, the last six contain material which is particular to the theory of
curves. We give a portion of the general theory of elliptic curves, the zeta
function of a curve and Riemann hypothesis. For most of the book I consider
irreducible varieties over an algebraically closed field. In Chapter 14, we work
over a finite field.
The most significant omission is intersection theory. Intersection theory is
vital to the understanding of surfaces. In the theory of curves, the theory
of correspondences on a curve X is just the intersection theory on the surface
X x X, which was put on a rigorous foundation by Well in his Foundations [30]
in preparation for proving the Riemann hypothesis for curves of higher genus.
Although the intersection theory is not covered here, the material in Chap-
ter 5 is very relevant to this topic. ! would like to point out Fulton's brief
and insightful introduction [9] to the intersection theory, which may be more
accessible than his definitive treatise [10].
I will maintain a web page at http://math. stanford. edu/~bump/ag. html
for this book. In partichlar, if errors are found, consult this web page for
corrections.
I would like to thank Sen Hu and E. H. Chionh of World Scientific for their
interest and encouragement, and for efficient editing of the manuscript.
- Hardcover: 250 pages
- Publisher: World Scientific Publishing Company (December 7, 1998)
- Language: English
- ISBN-10: 9810235615
- ISBN-13: 978-9810235611
- Product Dimensions: 8.8 x 6.3 x 0.7 inches
In 1989-1990 I taught a course in Algebraic Geometry at Stanford Univer-
sity, writing up lecture notes. These were revised for publication in 1998. In
1989-90 I covered the materiM in Chapters 1-14 in two quarters, and con-
tinued with a quarter on cohomology of coherent sheaves, lecturing out of
Hartshorne's book.
The aim is to make this a text that can be used in a one year at the graduate
level. I have tried to give complete proofs assuming a background in algebra at
the level one expects from a first or second year graduate student. The point
of view here is that of Serre [23] or Chapter I of Mumford [21]--a variety
is a ringed space locally isomorphic to an affine variety over a field, which is
algebraically closed except in Chapter 14. Although I do not treat schemes I
trust the reader will not find the transition too difficult.
The first eight sections contain material applicable to varieties of every
dimension, the last six contain material which is particular to the theory of
curves. We give a portion of the general theory of elliptic curves, the zeta
function of a curve and Riemann hypothesis. For most of the book I consider
irreducible varieties over an algebraically closed field. In Chapter 14, we work
over a finite field.
The most significant omission is intersection theory. Intersection theory is
vital to the understanding of surfaces. In the theory of curves, the theory
of correspondences on a curve X is just the intersection theory on the surface
X x X, which was put on a rigorous foundation by Well in his Foundations [30]
in preparation for proving the Riemann hypothesis for curves of higher genus.
Although the intersection theory is not covered here, the material in Chap-
ter 5 is very relevant to this topic. ! would like to point out Fulton's brief
and insightful introduction [9] to the intersection theory, which may be more
accessible than his definitive treatise [10].
I will maintain a web page at http://math. stanford. edu/~bump/ag. html
for this book. In partichlar, if errors are found, consult this web page for
corrections.
I would like to thank Sen Hu and E. H. Chionh of World Scientific for their
interest and encouragement, and for efficient editing of the manuscript.
