'''Algebraic geometry''' is a branch of [[mathematics]] which, as the name suggests, combines techniques of [[abstract algebra]], especially [[commutative algebra]], with the language and the problematics of [[geometry]]. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as [[complex analysis]], [[topology]] and [[number theory]]. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where [[equation solving]] leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

The fundamental objects of study in algebraic geometry are [[algebraic variety|'''algebraic varieties''']], geometric manifestations of [[solution set|solutions]] of systems of [[polynomial|polynomial equations]]. [[Plane algebraic curve]]s, which include [[line (mathematics)|lines]], [[circle]]s, [[parabola]]s, [[lemniscate]]s, and [[Cassini oval]]s, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.

[[Descartes]]'s idea of [[coordinates]] is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of [[real number]]s, but first [[complex number]]s, and then elements of an arbitrary [[field (mathematics)|field]] became acceptable. [[Homogeneous coordinates]] of [[projective geometry]] offered an extension of the notion of coordinate system in different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in [[topology]] and [[complex geometry]].

One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by [[Grothendieck]] and [[Jean-Pierre Serre|Serre]], is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a [[regular function]] and a [[regular map]] and extensively draws on [[sheaf theory]]. Another important difference lies in the scope of the subject. Grothendieck's idea of [[scheme (mathematics)|'''scheme''']] provides the language and the tools for geometric treatment
of arbitrary [[commutative ring]]s and, in particular, bridges algebraic geometry with [[algebraic number theory]]. [[Andrew Wiles]]'s celebrated [[modularity theorem|proof]] of [[Fermat's last theorem]] is a vivid testament to the power of this approach. [[André Weil]], Grothendieck, and [[Pierre Deligne|Deligne]] also [[Weil cohomology theory|demonstrated]] that the fundamental ideas of topology of [[smooth manifold|manifolds]] have deep analogues in algebraic geometry over [[finite field]]s.

== Zeros of simultaneous polynomials ==
[[Image:Slanted_circle.png|thumb|200px|right|Sphere and slanted circle]]
In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of [[polynomial]]s, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional [[sphere]] in three-dimensional [[Euclidean space]] '''R'''³ could be defined as the set of all points (''x'',''y'',''z'') with

:<math>x^2+y^2+z^2-1=0.\,</math>

A "slanted" circle in '''R'''³ can be defined as the set of all points (''x'',''y'',''z'') which satisfy the two polynomial equations

:<math>x^2+y^2+z^2-1=0,\,</math>
:<math>x+y+z=0.\,</math>

== Affine varieties ==
First we start with a [[field (mathematics)|field]] ''k''. In classical algebraic geometry, this field was always the complex numbers '''C''', but many of the same results are true if we assume only that ''k'' is [[algebraically closed field|algebraically closed]]. We define '''A'''ⁿ(''k'') (or more simply '''A'''^''n'', when ''k'' is clear from the context), called the '''affine n-space over k''', to be ''k''^''n''. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that ''k''ⁿ carries. Abstractly speaking, '''A'''^''n'' is, for the moment, just a collection of points.

A function ''f'' : '''A'''^''n'' → '''A'''¹ is said to be '''regular''' if it can be written as a polynomial, that is, if there is a polynomial ''p'' in ''k''[''x''₁,...,''x''_''n''] such that ''f''(''t''₁,...,''t''_''n'') = ''p''(''t''₁,...,''t''_''n'') for every point (''t''₁,...,''t''_''n'') of '''A'''^''n''.

Regular functions on affine ''n''-space are thus exactly the same as polynomials over ''k'' in ''n'' variables. We will write the regular functions on '''A'''^''n'' as ''k''['''A'''^''n''].

We say that a polynomial ''vanishes'' at a point if evaluating it at that point gives zero. Let ''S'' be a set of polynomials in ''k''['''A'''ⁿ]. The ''vanishing set of S'' (or ''vanishing locus'') is the set ''V''(''S'') of all points in '''A'''^''n'' where every polynomial in ''S'' vanishes. In other words,

:<math>V(S) = \{(t_1,\dots,t_n)|\forall p\in S, p(t_1,\dots,t_n) = 0\}.\,</math>

A subset of '''A'''^''n'' which is ''V''(''S''), for some ''S'', is called an '''algebraic set'''. The ''V'' stands for ''variety'' (a specific type of algebraic set to be defined below).

Given a subset ''U'' of '''A'''^''n'', can one recover the set of polynomials which generate it? If ''U'' is ''any'' subset of '''A'''^''n'', define ''I''(''U'') to be the set of all polynomials whose vanishing set contains ''U''. The ''I'' stands for [[ideal (ring theory)|ideal]]: if two polynomials ''f'' and ''g'' both vanish on ''U'', then ''f''+''g'' vanishes on ''U'', and if ''h'' is any polynomial, then ''hf'' vanishes on ''U'', so ''I''(''U'') is always an ideal of ''k''['''A'''^''n''].

Two natural questions to ask are:
* Given a subset ''U'' of '''A'''^''n'', when is ''U'' = ''V''(''I''(''U''))?
* Given a set ''S'' of polynomials, when is ''S'' = ''I''(''V''(''S''))?

The answer to the first question is provided by introducing the [[Zariski topology]], a topology on '''A'''^''n'' which directly reflects the algebraic structure of ''k''['''A'''^''n'']. Then ''U'' = ''V''(''I''(''U'')) if and only if ''U'' is a Zariski-closed set. The answer to the second question is given by [[Hilbert's Nullstellensatz]]. In one of its forms, it says that ''I''(''V''(''S'')) is the [[prime radical]] of the ideal generated by ''S''. In more abstract language, there is a [[Galois connection]], giving rise to two [[closure operator]]s; they can be identified, and naturally play a basic role in the theory; the [[Galois_connection#Examples|example]] is elaborated at Galois connection.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set ''U''. [[Hilbert's basis theorem]] implies that ideals in ''k''['''A'''^''n''] are always finitely generated.

An algebraic set is called '''[[irreducible component|irreducible]]''' if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a '''[[algebraic variety|variety]]'''. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a [[prime ideal]] of the polynomial ring.

== Regular functions ==

Just as [[continuous function]]s are the natural maps on [[topological space]]s and [[smooth function]]s are the natural maps on [[differentiable manifold]]s, there is a natural class of functions on an algebraic set, called regular functions. A '''regular function''' on an algebraic set ''V'' contained in '''A'''ⁿ is defined to be the restriction of a regular function on '''A'''ⁿ, in the sense we defined above.

It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a [[normal space|normal]] [[topological space]], where the [[Tietze extension theorem]] guarantees that a continuous function on a closed subset always extends to the ambient topological space.

Just as with the regular functions on affine space, the regular functions on ''V'' form a ring, which we denote by ''k''[''V'']. This ring is called the '''coordinate ring of ''V'''''.

Since regular functions on V come from regular functions on '''A'''ⁿ, there should be a relationship between their coordinate rings. Specifically, to get a function in ''k''[''V''] we took a function in ''k''['''A'''ⁿ], and we said that it was the same as another function if they gave the same values when evaluated on ''V''. This is the same as saying that their difference is zero on V. From this we can see that ''k''[''V''] is the quotient ''k''['''A'''ⁿ]/I(''V'').

== The category of affine varieties ==

Using regular functions from an affine variety to '''A'''¹, we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let ''V'' be a variety contained in '''A'''ⁿ. Choose ''m'' regular functions on ''V'', and call them ''f''₁, ..., ''f''_''m''. We define a '''regular function''' ''f'' from ''V'' to '''A'''^m by letting ''f''(''t''₁, ..., ''t''_''n'') = (''f''₁, ..., ''f''_''m''). In other words, each ''f''_''i'' determines one coordinate of the [[range (mathematics)|range]] of ''f''.

If ''V''<nowiki>'</nowiki> is a variety contained in '''A'''^m, we say that ''f'' is a '''regular function''' from ''V'' to ''V''<nowiki>'</nowiki> if the range of ''f'' is contained in ''V''<nowiki>'</nowiki>.

This makes the collection of all affine varieties into a [[category theory|category]], where the objects are affine varieties and the [[morphism]]s are regular maps. The following theorem characterizes the category of affine varieties:

: The category of affine varieties is the [[dual (category theory)|opposite category]] to the category of finitely generated [[integral domain|integral]] ''k''-[[algebra over a field|algebras]] and their homomorphisms.

== Projective space ==
[[Image:Parabola in projective space.png|thumb|300px|parabola (y=x², blue) and cubic (y=x³, red) in projective space]]
Consider the variety ''V''(''y''&nbsp;-&nbsp;''x''²). If we draw it, we get a [[parabola]]. As ''x'' increases, the slope of the line from the origin to the point (''x'',&nbsp;''x''²) becomes larger and larger. As ''x'' decreases, the slope of the same line becomes smaller and smaller.

Compare this to the variety ''V''(''y''&nbsp;-&nbsp;''x''³). This is a [[cubic equation]]. As ''x'' increases, the slope of the line from the origin to the point (''x'',&nbsp;''x''³) becomes larger and larger just as before. But unlike before, as ''x'' decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y-x³) is different from the behavior "at infinity" of ''V''(''y''&nbsp;-&nbsp;''x''²). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space.

The remedy to this is to work in [[projective space]]. Projective space has properties analogous to those of a [[compact space|compact]] [[Hausdorff space]]. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, ''V''(''y''&nbsp;-&nbsp;''x''³) has a [[mathematical singularity|singularity]] at one of those extra points, but ''V''(''y''&nbsp;-&nbsp;''x''²) is smooth.

While [[projective geometry]] was originally established on a [[synthetic geometry|synthetic]] foundation, the use of [[homogeneous coordinates]] allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, [[Bézout's theorem]] on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry.

== The modern viewpoint ==
The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in [[Alexander Grothendieck]]'s concept of a [[scheme (mathematics)|scheme]]. Schemes start with the observation that if finitely generated reduced k-algebras are geometrical objects, then perhaps arbitrary commutative rings should also be geometrical objects. As such, schemes become both a more general algebro-geometric object, and a convenient language to describe those objects. This language of schemes has proved to be a valuable way of dealing with geometric concepts and has become a cornerstone of modern algebraic geometry.

== History ==
{{disputed}}
===Prehistory: Before the 19th century===
Some of the roots of algebraic geometry date back to the later work of the [[Hellenistic Greece|Hellenistic Greeks]] such as [[Archimedes]] and [[Apollonius of Perga|Apollonius]] on [[conic sections]].<ref>Kline, M. (1972) ''Mathematical Thought from Ancient to Modern Times'' (Volume 1). Oxford University Press. pp. 108, 90.</ref> The geometrical methods of the Greeks were first applied in a recognizably algebraic setting by 10th century scholar, Abu’l Wafa al-Buznaji, an astronomer and mathematician, who wrote a book of applied geometry called ''Kitab al handasa'' or the Book of Geometry. In it he provided solutions of geometrical problems with one opening of the compass; constructions of a square equivalent to other squares; regular polyhedra; approximate construction of regular heptagon (taking for its side half the side of the equilateral triangle inscribed in the same circle); constructions of parabola by points; geometrical solution of x4 = a and x4 + ax3 = b. Then [[Muhammad ibn Jābir al-Harrānī al-Battānī|Muhammed Al Battani]], a 10th century mathematician and astronomer from Baghdad , computed sine, tangent and cotangent tables from 0° to 90° with great accuracy. Last and not least [[Persian people|Persian]] mathematician [[Omar Khayyám]] (born 1048 A.D.) discovered the general method of solving [[cubic equation]]s by intersecting a parabola with a circle.<ref>Kline ''ibid'', pp. 193-195.</ref> Each of these primordial developments in algebraic geometry dealt with questions of finding and describing the intersections of algebraic curves.

Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of [[Renaissance]] mathematicians such as [[Gerolamo Cardano]] and [[Niccolò Fontana Tartaglia]] on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably [[Blaise Pascal]] who argued against the use of algebraic and analytical methods in geometry.<ref>Kline ''ibid'', p. 279.</ref> The French mathematicians [[Franciscus Vieta]] and later [[René Descartes]] and [[Pierre de Fermat]] revolutionized the conventional way of thinking about construction problems through the introduction of [[coordinate geometry]]. They were interested primarily in the properties of ''algebraic curves'', such as those defined by [[Diophantine equations]] (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).

During the same period, Blaise Pascal and [[Gérard Desargues]] approached geometry from a different perspective, developing the [[synthetic geometry|synthetic]] notions of [[projective geometry]]. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ''ruler and compass construction''. Ultimately, the [[analytic geometry]] of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]]. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the ''calculus of infinitesimals'' of [[Joseph Louis Lagrange]] and [[Leonard Euler]].

===Nineteenth and early 20th century===
It took the simultaneous 19th century developments of [[non-Euclidean geometry]] and [[Abelian integral]]s in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by [[Edmond Laguerre]] and [[Arthur Cayley]], who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of ''homogeneous polynomial forms'', and more specifically [[quadratic form]]s, on projective space. Subsequently, [[Felix Klein]] studied projective geometry (along with other sorts of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of [[transformation group|transformations]] on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental [[Kleinian geometry]] on projective space, they concerned themselves also with the higher degree [[birational transformation]]s. This weaker notion of congruence would later lead members of the 20th century [[Italian school of algebraic geometry]] to classify [[algebraic surface]]s up to [[birational isomorphism]].

The second early 19th century development, that of Abelian integrals, would lead [[Bernhard Riemann]] to the development of Riemann surfaces. <!--Continue later. Intermediate save. -->

===Twentieth century===
[[van der Waerden]], [[Oscar Zariski]], [[André Weil]] and others attempted to develop a rigorous foundation for algebraic geometry based on contemporary [[commutative algebra]], including [[valuation theory]] and the theory of [[ideal]]s.

In the 1950s and 1960s [[Jean-Pierre Serre]] and [[Alexander Grothendieck]] recast the foundations making use of [[sheaf theory]]. Later, from about 1960, the idea of [[scheme (mathematics)|schemes]] was worked out, in conjunction with a very refined apparatus of [[homological algebra|homological techniques]]. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to [[number theory]] and to more classical geometric questions on algebraic varieties, [[singularity theory|singularities]] and [[moduli space|moduli]].

An important class of varieties, not easily understood directly from their defining equations, are the [[abelian variety|abelian varieties]], which are the projective varieties whose points form an abelian [[group (mathematics)|group]].
The prototypical examples are the [[elliptic curve]]s, which have a rich theory. They were instrumental in the proof of [[Fermat's last theorem]] and are also used in [[elliptic curve cryptography]].

While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of [[Gröbner basis|Gröbner bases]] which is employed in all [[computer algebra]] systems.

==See also==
* [[Geometric algebra]]
* [[List of publications in mathematics#Algebraic geometry|Important publications in algebraic geometry]]
* [[List of algebraic surfaces]]
* [[Root-finding algorithm]]

==Notes==
<references/>

==References==
A classical textbook, predating schemes:

*{{cite book
| author = [[W. V. D. Hodge]]
| coauthors = [[Daniel Pedoe]]
| year = 1994
| title = Methods of Algebraic Geometry: Volume 1
| publisher = [[Cambridge University Press]]
| id = ISBN 0-521-46900-7
}}
*{{cite book
| author = [[W. V. D. Hodge|Hodge, W. V. D.]]
| coauthors = [[Daniel Pedoe|Pedoe, Daniel]]
| year = 1994
| title = Methods of Algebraic Geometry: Volume 2
| publisher = [[Cambridge University Press]]
| id = ISBN 0-521-46901-5
}}
*{{cite book
| author = [[W. V. D. Hodge|Hodge, W. V. D.]]
| coauthors = [[Daniel Pedoe|Pedoe, Daniel]]
| year = 1994
| title = Methods of Algebraic Geometry: Volume 3
| publisher = [[Cambridge University Press]]
| id = ISBN 0-521-46775-6
}}

Modern textbooks that do not use the language of schemes:

*{{cite book
| author = [[David Cox]]
| coauthors = [[John Little, Donal O'Shea]]
| year = 1997
| title = Ideals, Varieties, and Algorithms
| edition = second edition
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-94680-2
}}
*{{cite book
| author = [[Phillip Griffiths]]
| coauthors = [[Joe Harris (mathematician)|Joe Harris]]
| year = 1994
| title = Principles of Algebraic Geometry
| publisher = Wiley-Interscience
| id = ISBN 0-471-05059-8
}}
*{{cite book
| author = [[Joe Harris (mathematician)|Joe Harris]]
| year = 1995
| title = Algebraic Geometry: A First Course
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-97716-3
}}
*{{cite book
| author = [[David Mumford]]
| year = 1995
| title = Algebraic Geometry I: Complex Projective Varieties
| edition = 2nd ed.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 3-540-58657-1
}}
*{{cite book
| author = [[Miles Reid]]
| year = 1988
| title = Undergraduate Algebraic Geometry
| publisher = Cambridge University Press
| id = ISBN 0-521-35662-8
}}
*{{cite book
| author = [[Igor Shafarevich]]
| year = 1995
| title = Basic Algebraic Geometry I: Varieties in Projective Space
| edition = 2nd ed.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-54812-2
}}

Textbooks and references for schemes:
*{{cite book
| author = [[David Eisenbud]]
| coauthors = [[Joe Harris (mathematician)|Joe Harris]]
| year = 1998
| title = The Geometry of Schemes
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-98637-5
}}
*{{cite book
| author = [[Alexander Grothendieck]]
| year = 1960
| title = [[Éléments de géométrie algébrique]]
| publisher = Publications mathématiques de l'[[IHÉS]]
}}
*{{cite book
| author = [[Alexander Grothendieck]]
| year = 1971
| title = [[Éléments de géométrie algébrique]]
| volume = 1
| edition = 2nd ed.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 3-540-05113-9
}}
*{{cite book
| author = [[Robin Hartshorne]]
| year = 1997
| title = [[Hartshorne%27s_Algebraic_Geometry|Algebraic Geometry]]
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-90244-9
}}
*{{cite book
| author = [[David Mumford]]
| year = 1999
| title = The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
| edition = 2nd ed.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 3-540-63293-X
}}
*{{cite book
| author = [[Igor Shafarevich]]
| year = 1995
| title = Basic Algebraic Geometry II: Schemes and Complex Manifolds
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| id = ISBN 0-387-54812-2
}}

On the Internet:

* Kevin R. Coombes: [http://odin.mdacc.tmc.edu/~krc/agathos/ ''Algebraic Geometry: A Total Hypertext Online System'']
* [http://planetmath.org/encyclopedia/AlgebraicGeometry.html ''Algebraic geometry''] entry on [http://planetmath.org/ PlanetMath]
* [http://eqworld.ipmnet.ru/en/solutions/ae.htm ''Algebraic Equations and Systems of Algebraic Equations''] at EqWorld: The World of Mathematical Equations

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