Algebraic Geometry I. From Algebraic Varieties to Schemes - K. Ueno
It has often been said that algebraic geometry is a difficult field
in mathematics. There certainly was a time when algebraic geom-
etry was a difficult geometry. In particular, the theory of algebraic
curves o f the Italian school fro m the late nineteenth century through
the first half of the twentieth century was indeed difficult. Intuitive
arguments proceeded without rigorous proofs. Legend has it that
one of the leaders of the Italian school, Enriques, once said, "It is a
nobleman's work to find theorems, and it is a slave's work to prove
them. Mathematicians are noblemen." Their sharpness of intuition
might well convince us of that legend, but it was nearly impossible
for common mathematicians to follow the arguments.
The plans to provide mathematically solid foundations for such
intuitionistic algebraic geometry were carried out by van der Waer
den, Zariski, Weil, Chevalley, and others, using abstract algebra as it
developed in the 1930's. Zariski and Weil provided a foundation for
algebraic geometry for their time period. Based on their foundation,
Weil was able to prove the Riemann hypothesi for an algebraic curve
defined over a fmite field, establishing the closely related theory of
abelJan varieties over a field of positive characteristic, and Zariski es-
tablished birational geometry over a field of arbitrary characteristic.
The theorems of Well and Zariski were among the main results of
their era.
- Paperback: 168 pages
- Publisher: American Mathematical Society (September 27, 1999)
- Language: English
- ISBN-10: 0821808621
- ISBN-13: 978-0821808627
- Product Dimensions: 8.4 x 5.3 x 0.4 inches
It has often been said that algebraic geometry is a difficult field
in mathematics. There certainly was a time when algebraic geom-
etry was a difficult geometry. In particular, the theory of algebraic
curves o f the Italian school fro m the late nineteenth century through
the first half of the twentieth century was indeed difficult. Intuitive
arguments proceeded without rigorous proofs. Legend has it that
one of the leaders of the Italian school, Enriques, once said, "It is a
nobleman's work to find theorems, and it is a slave's work to prove
them. Mathematicians are noblemen." Their sharpness of intuition
might well convince us of that legend, but it was nearly impossible
for common mathematicians to follow the arguments.
The plans to provide mathematically solid foundations for such
intuitionistic algebraic geometry were carried out by van der Waer
den, Zariski, Weil, Chevalley, and others, using abstract algebra as it
developed in the 1930's. Zariski and Weil provided a foundation for
algebraic geometry for their time period. Based on their foundation,
Weil was able to prove the Riemann hypothesi for an algebraic curve
defined over a fmite field, establishing the closely related theory of
abelJan varieties over a field of positive characteristic, and Zariski es-
tablished birational geometry over a field of arbitrary characteristic.
The theorems of Well and Zariski were among the main results of
their era.
