''Algebraic Geometry'' is an
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...
written by
Robin Hartshorne
__NOTOC__
Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry.
Career
Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
and published by
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
in 1977.
MathSciNet
MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...
lists more than 2500 citations of this book.
Importance
It was the first extended treatment of
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
written as a text intended to be accessible to graduate students.
Contents
The first chapter, titled "Varieties", deals with the classical algebraic geometry of
varieties
Variety may refer to:
Arts and entertainment Entertainment formats
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Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
over algebraically closed fields. This chapter uses many classical results in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, including
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.
Notes
References
*
* {{citation, last=Shatz, first=Stephen S., title=Review: Robin Hartshorne, ''Algebraic geometry'', journal=Bull. Amer. Math. Soc. (N.S.), volume=1, issue=3, year=1979, pages=553–560, url=http://projecteuclid.org/euclid.bams/1183544340, doi = 10.1090/S0273-0979-1979-14618-4, doi-access=free
Mathematics textbooks
1977 non-fiction books
Algebraic geometry
Monographs
Springer Science+Business Media books