''Foundations of Algebraic Geometry'' is a book by that develops

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

over

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

s of any characteristic. In particular it gives a careful treatment of

intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...

by defining the local intersection multiplicity of two subvarieties. Weil was motivated by the need for a rigorous theory of correspondences on

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s in positive characteristic, which he used in his proof of the

Riemann hypothesis for curves over finite fields In mathematics, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^k\right) where is a non-singular -dimensional projective algebr ...

. Weil introduced abstract rather than

projective varieties In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the ...

partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective. In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by

Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

's development of

scheme Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'', a BBC Scotland documentary TV series * The Scheme (band), an English pop band * ''The Scheme'', an action role-playing video game for the PC-8801, made by Quest Corporation * ...

References

* * * * *{{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , title=Book Review: Foundations of algebraic geometry , doi=10.1090/S0002-9904-1948-09040-1 , mr=1565074 , year=1948 , journal=

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

, volume=54 , issue=7 , pages=671–675, bibcode=1948Sci...107...75W , doi-access=free

External links

Extracts from the preface of ''Foundations of Algebraic Geometry''
1946 non-fiction books Algebraic geometry Mathematics books History of mathematics

See also

References

External links