TheInfoList

Algebraic geometry is a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, classically studying zeros of
multivariate polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. Modern algebraic geometry is based on the use of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
ic techniques, mainly from
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, for solving about these sets of zeros. The fundamental objects of study in algebraic geometry are
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the us ...
, which are geometric manifestations of
solutions Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent. In chemistry, a solution is a special type of Homogeneous and ...

of
systems of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...
. Examples of the most studied classes of algebraic varieties are:
plane algebraic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, which include
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA, sever ...

,
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

s,
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

s,
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

s,
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

s,
cubic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s like
elliptic curve In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
s, and quartic curves like
lemniscate 400px, The lemniscate of Bernoulli and its two foci In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the us ...

s and
Cassini oval A Cassini oval is a quartic plane curve defined as the Set (mathematics), set (or Locus (mathematics), locus) of points in the plane (mathematics), plane such that the product of the distances to two fixed points is constant. This may be contrasted ...

s. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Basic questions involve the study of the points of special interest like the singular points, the
inflection point . In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature In mathematics, curvature is any of se ...

s and the
points at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
. More advanced questions involve the
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

of the curve and relations between the curves given by different equations. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
,
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. Initially a study of
systems of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...
in several variables, the subject of algebraic geometry starts where
equation solving alt=Illustration of Newton's method, An example of using Newton–Raphson method to solve numerically the equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ma ...
leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique. In the 20th century, algebraic geometry split into several subareas. * The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. *
Real algebraic geometryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is the study of the real points of an algebraic variety. *
Diophantine geometry In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be wr ...
and, more generally,
arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...
is the study of the points of an algebraic variety with coordinates in
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 grassl ...
s that are not
algebraically closed In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and occur in
algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
, such as the field of
rational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s,
number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s,
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, function fields, and ''p''-adic fields. * A large part of
singularity theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is devoted to the singularities of algebraic varieties. * Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and
computer algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, with the rise of computers. It consists mainly of
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

design and
software Software is a collection of instructions Instruction or instructions may refer to: Computing * Instruction, one operation of a processor within a computer architecture instruction set * Computer program, a collection of instructions Music * I ...

development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an 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 s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, differential and
complex geometry In mathematics, complex geometry is the study of complex manifolds, Complex algebraic variety, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this cate ...
. One key achievement of this abstract algebraic geometry is
Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such top ...
's
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 multiplicity (mathematics), multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebra ...
which allows one to use
sheaf theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz ' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

, with a
maximal ideal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the
coordinate ring In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called
Fermat's last theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
is an example of the power of this approach.

# Basic notions

## Zeros of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, meaning the set of all points that simultaneously satisfy one or more
polynomial equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For instance, the
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
R3 could be defined as the set of all points (''x'',''y'',''z'') with :$x^2+y^2+z^2-1=0.\,$ A "slanted" circle in R3 can be defined as the set of all points (''x'',''y'',''z'') which satisfy the two polynomial equations :$x^2+y^2+z^2-1=0,\,$ :$x+y+z=0.\,$

## Affine varieties

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 grassl ...
''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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. We consider the
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
of dimension ''n'' over ''k'', denoted An(''k'') (or more simply A''n'', when ''k'' is clear from the context). When one fixes a coordinate system, one may identify An(''k'') with ''k''''n''. The purpose of not working with ''k''''n'' is to emphasize that one "forgets" the vector space structure that ''k''n carries. A function ''f'' : A''n'' → A1 is said to be ''polynomial'' (or ''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''(''M'') = ''p''(''t''1,...,''t''''n'') for every point ''M'' with coordinates (''t''1,...,''t''''n'') in A''n''. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in A''n''. When a coordinate system is chosen, the regular functions on the affine ''n''-space may be identified with the ring of
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
s in ''n'' variables over ''k''. Therefore, the set of the regular functions on A''n'' is a ring, which is denoted ''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'' ''An The ''vanishing set of S'' (or ''vanishing locus'' or ''zero set'') is the set ''V''(''S'') of all points in A''n'' where every polynomial in ''S'' vanishes. Symbolically, :$V\left(S\right) = \.\,$ 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 Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
: 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 the polynomial ring ''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 In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...
, a topology on A''n'' whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of ''k'' ''A''n'' Then ''U'' = ''V''(''I''(''U'')) if and only if ''U'' is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz ' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

. In one of its forms, it says that ''I''(''V''(''S'')) is the
of the ideal generated by ''S''. In more abstract language, there is a
Galois connection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
, giving rise to two
closure operatorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s; they can be identified, and naturally play a basic role in the theory; the
example Example may refer to: * ''exempli gratia Notes and references Notes References Sources * * * Further reading * * {{Latin phrases E ...'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain na ...
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 In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. Statement If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. David H ...
implies that ideals in ''k'' ''A''n''are always finitely generated. An algebraic set is called '' irreducible'' if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the ''irreducible components'' of the algebraic set. An irreducible algebraic set is also called a ''
variety Variety may refer to: Science and technology Mathematics * Algebraic variety, the set of solutions of a system of polynomial equations * Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra Hort ...
''. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use ''irreducible variety'' to make the distinction when needed.

## Regular functions

Just as
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are the natural maps on
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s and
smooth function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

s are the natural maps on
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
s, there is a natural class of functions on an algebraic set, called ''regular functions'' or ''polynomial functions''. A regular function on an algebraic set ''V'' contained in A''n'' is the restriction to ''V'' of a regular function on A''n''. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic. 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
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, where the
Tietze extension theorem In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
of V''. Since regular functions on V come from regular functions on A''n'', there is a relationship between the coordinate rings. Specifically, if a regular function on ''V'' is the restriction of two functions ''f'' and ''g'' in ''k'' ''A''n'' then ''f'' − ''g'' is a polynomial function which is null on ''V'' and thus belongs to ''I''(''V''). Thus ''k'' 'V''may be identified with ''k'' ''A''n''''I''(''V'').

## Morphism of affine varieties

Using regular functions from an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map 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 map'' ''f'' from ''V'' to A''m'' by letting . In other words, each ''f''''i'' determines one coordinate of the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of ''f''. If ''V''′ is a variety contained in A''m'', we say that ''f'' is a ''regular map'' from ''V'' to ''V''′ if the range of ''f'' is contained in ''V''′. The definition of the regular maps apply also to algebraic sets. The regular maps are also called ''morphisms'', as they make the collection of all affine algebraic sets into a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
, where the objects are the affine algebraic sets and the
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. Given a regular map ''g'' from ''V'' to ''V''′ and a regular function ''f'' of ''k'' 'V''′ then . The map is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
from ''k'' 'V''′to ''k'' 'V'' Conversely, every ring homomorphism from ''k'' 'V''′to ''k'' 'V''defines a regular map from ''V'' to ''V''′. This defines an
equivalence of categories In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
between the category of algebraic sets and the
opposite category In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
of the finitely generated reduced ''k''-algebras. This equivalence is one of the starting points 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 multiplicity (mathematics), multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebra ...
.

## Rational function and birational equivalence

In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. If ''V'' is an affine variety, its coordinate ring is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
and has thus a
field of fractions In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
which is denoted ''k''(''V'') and called the ''field of the rational functions'' on ''V'' or, shortly, the '' function field'' of ''V''. Its elements are the restrictions to ''V'' of the
rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s over the affine space containing ''V''. The
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of a rational function ''f'' is not ''V'' but the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of the subvariety (a hypersurface) where the denominator of ''f'' vanishes. As with regular maps, one may define a ''rational map'' from a variety ''V'' to a variety ''V'''. As with the regular maps, the rational maps from ''V'' to ''V''' may be identified to the field homomorphisms from ''k''(''V''') to ''k''(''V''). Two affine varieties are ''birationally equivalent'' if there are two rational functions between them which are
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a ''
rational varietyIn mathematics, a rational variety is an algebraic variety, over a given field (mathematics), field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field of an algebraic variety ...
'' if it is birationally equivalent to an affine space. This means that the variety admits a ''rational parameterization'', that is a parametrization with
rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s. For example, the circle of equation $x^2+y^2-1=0$ is a rational curve, as it has the
parametric equation In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...
:$x=\frac$ :$y=\frac\,,$ which may also be viewed as a rational map from the line to the circle. The problem of
resolution of singularities In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commu ...
is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also smooth completion). It was solved in the affirmative in characteristic 0 by
Heisuke Hironaka is a Japanese Japanese may refer to: * Something from or related to Japan , image_flag = Flag of Japan.svg , alt_flag = Centered deep red circle on a white rectangle , image_coat = ...
in 1964 and is yet unsolved in finite characteristic.

## Projective variety

Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number ''i'', a root of the polynomial , projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet. To see how this might come about, consider the variety . If we draw it, we get a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''2) also goes to positive infinity. As ''x'' goes to negative infinity, the slope of the same line goes to negative infinity. Compare this to the variety ''V''(''y'' − ''x''3). This is a
cubic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''3) goes to positive infinity just as before. But unlike before, as ''x'' goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of ''V''(''y'' − ''x''3) is different from the behavior "at infinity" of ''V''(''y'' − ''x''2). The consideration of the ''projective completion'' of the two curves, which is their prolongation "at infinity" in the
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the
line at infinity In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

, while the point at infinity of the cubic curve is a
cusp Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. Also, both curves are rational, as they are parameterized by ''x'', and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example,
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zero of a function, zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the p ...
on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. Nowadays, the ''
projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
'' P''n'' of dimension ''n'' is usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimension , or equivalently to the set of the vector lines in a vector space of dimension . When a coordinate system has been chosen in the space of dimension , all the points of a line have the same set of coordinates, up to the multiplication by an element of ''k''. This defines the
homogeneous coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of a point of P''n'' as a sequence of elements of the base field ''k'', defined up to the multiplication by a nonzero element of ''k'' (the same for the whole sequence). A polynomial in variables vanishes at all points of a line passing through the origin if and only if it is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about th ...
. In this case, one says that the polynomial ''vanishes'' at the corresponding point of P''n''. This allows us to define a ''projective algebraic set'' in P''n'' as the set , where a finite set of homogeneous polynomials vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced
homogeneous ideal Homogeneity and heterogeneity are concepts often used in the Science, sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform in ...
s which define them. The ''projective varieties'' are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose
homogeneous coordinate ring In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), a ...
is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, the ''projective coordinates ring'' being defined as the quotient of the graded ring or the polynomials in variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, the ''field of the rational functions'' or ''function field '' is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.

# Real algebraic geometry

Real algebraic geometry is the study of the real points of algebraic varieties. The fact that the field of the real numbers is an
ordered field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
cannot be ignored in such a study. For example, the curve of equation $x^2+y^2-a=0$ is a circle if $a>0$, but does not have any real point if $a<0$. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the ''semi-algebraic sets'', which are the solutions of systems of polynomial equations and polynomial inequalities. For example, a branch of the
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

of equation $x y-1 = 0$ is not an algebraic variety, but is a semi-algebraic set defined by $x y-1=0$ and $x>0$ or by $x y-1=0$ and $x+y>0$. One of the challenging problems of real algebraic geometry is the unsolved
Hilbert's sixteenth problemHilbert's 16th problem was posed by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and de ...
: Decide which respective positions are possible for the ovals of a nonsingular
plane curve In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought ...
of degree 8.

# Computational algebraic geometry

One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at
Marseille Marseille ( , , ; also spelled in English as Marseilles; oc, Marselha ) is the prefecture A prefecture (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European langua ...

, France in June 1979. At this meeting, * Dennis S. Arnon showed that
George E. CollinsGeorge E. Collins (January 10, 1928 in Stuart, Iowa Stuart is a city in Lincoln Township, Adair County, and in Stuart Township, Guthrie County, Iowa, Guthrie County, in the U.S. state of Iowa. That part of the city within Guthrie County is part o ...
's
Cylindrical algebraic decomposition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(CAD) allows the computation of the topology of semi-algebraic sets, *
Bruno Buchberger Bruno Buchberger (born 22 October 1942) is Professor of Computer Mathematics at Johannes Kepler University in Linz Linz (; ; cs, Linec) is the capital of Upper Austria and List of cities and towns in Austria, third-largest city in Austria. In ...

presented the Gröbner bases and his algorithm to compute them, *
Daniel Lazard Daniel Lazard (born December 10, 1941) is a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number t ...
presented a new algorithm for solving systems of homogeneous polynomial equations with a computational complexity which is essentially polynomial in the expected number of solutions and thus simply exponential in the number of the unknowns. This algorithm is strongly related with Macaulay's
multivariate resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root of a function, root (possibly in a field extension), or, equivalently, a co ...
. Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables. A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades. The main computational method is homotopy continuation. This supports, for example, a model of
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
computation for solving problems of algebraic geometry.

## Gröbner basis

A
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of Ideal (ring theory)#Ideal generated by a set, generating set of an ideal in ...
is a system of generators of a polynomial
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal. Given an ideal ''I'' defining an algebraic set ''V'': * ''V'' is empty (over an algebraically closed extension of the basis field), if and only if the Gröbner basis for any
monomial order In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (Monic polynomial, monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * ...
ing is reduced to . * By means of the
Hilbert seriesIn commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded algebra, graded commutative algebra finitely generated over a field (mathematics), field are three strongly related notions which measure the gro ...
one may compute the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
and the
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
of ''V'' from any Gröbner basis of ''I'' for a monomial ordering refining the total degree. * If the dimension of ''V'' is 0, one may compute the points (finite in number) of ''V'' from any Gröbner basis of ''I'' (see
Systems of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...
). * A Gröbner basis computation allows one to remove from ''V'' all irreducible components which are contained in a given hypersurface. * A Gröbner basis computation allows one to compute the Zariski closure of the image of ''V'' by the projection on the ''k'' first coordinates, and the subset of the image where the projection is not proper. * More generally Gröbner basis computations allow one to compute the Zariski closure of the image and the critical points of a rational function of ''V'' into another affine variety. Gröbner basis computations do not allow one to compute directly the primary decomposition of ''I'' nor the prime ideals defining the irreducible components of ''V'', but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases use
regular chainIn computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of Wu's method of characteristic set, characteristic set. Introduction Given a System of linear equ ...
s but may need Gröbner bases in some exceptional situations. Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply. Faugère F5 algorithm realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.

CAD is an algorithm which was introduced in 1973 by G. Collins to implement with an acceptable complexity the
Tarski–Seidenberg theoremIn mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by Polynomial#Polynomial equations, polynomial equations and inequality (mathematics), inequalities can be projected down onto ''n''- ...
on quantifier elimination over the real numbers. This theorem concerns the formulas of the
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
whose
atomic formula In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
s are polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operators ''and'' (∧), ''or'' (∨), ''not'' (¬), ''for all'' (∀) and ''exists'' (∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifier (∀, ∃). The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets. While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables. Since 1973, most of the research on this subject is devoted either to improve CAD or to find alternative algorithms in special cases of general interest. As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.

## Asymptotic complexity vs. practical efficiency

The basic general algorithms of computational geometry have a double exponential worst case
complexity Complexity characterises the behaviour of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environme ...
. More precisely, if ''d'' is the maximal degree of the input polynomials and ''n'' the number of variables, their complexity is at most $d^$ for some constant ''c'', and, for some inputs, the complexity is at least $d^$ for another constant ''c''′. During the last 20 years of the 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexity $d^$. Among these algorithms which solve a sub problem of the problems solved by Gröbner bases, one may cite ''testing if an affine variety is empty'' and ''solving nonhomogeneous polynomial systems which have a finite number of solutions.'' Such algorithms are rarely implemented because, on most entries Faugère's F4 and F5 algorithms have a better practical efficiency and probably a similar or better complexity (''probably'' because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases). The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite ''counting the number of connected components'', ''testing if two points are in the same components'' or ''computing a
Whitney stratificationIn differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A stratification of a topological space is a finite filtration (mathematics), fi ...
of a real algebraic set''. They have a complexity of $d^$, but the constant involved by ''O'' notation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency.

# Abstract modern viewpoint

The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes,
ind-scheme In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), a ...
s,
algebraic spaceIn mathematics, algebraic spaces form a generalization of the scheme (mathematics), schemes of algebraic geometry, introduced by for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topol ...
s,
algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or Scheme (mathematics), schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, s ...
s and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions. Most remarkably, in the late 1950s, algebraic varieties were subsumed into
Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

's concept of a scheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field ''k'', and the category of finitely generated reduced ''k''-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a
Grothendieck topologyIn category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is call ...
. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including
Nisnevich topologyIn algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of scheme (mathematics), schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the th ...
. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks. Sometimes other algebraic sites replace the category of affine schemes. For example,
Nikolai Durov Nikolai Valeryevich Durov (russian: Никола́й Вале́рьевич Ду́ров; born 21 November 1980) is a Russian programmer A computer programmer, sometimes called a software developer, a programmer or more recently a coder (esp ...
has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an
absolute geometry Absolute geometry is a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with pro ...
over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup. Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry. The term ''variety of algebras'' should not be confused with ''algebraic variety''. The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the
Grothendieck site In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
of affine schemes to a higher categorical site of derived affine schemes, by replacing the commutative rings with an infinity category of differential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). Quillen model categories, Segal categories and quasicategory, quasicategories are some of the most often used tools to formalize this yielding the ''derived algebraic geometry'', introduced by the school of Carlos Simpson, including Andre Hirschowitz, Bertrand Toën, Gabrielle Vezzosi, Michel Vaquié and others; and developed further by Jacob Lurie, Bertrand Toën, and Gabrielle Vezzosi. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from the early 1990s by Maxim Kontsevich and followers.

# History

## Before the 16th century

Some of the roots of algebraic geometry date back to the work of the Hellenistic Greece, Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length ''x'' so that the cube of side ''x'' contained the same volume as the rectangular box ''a''2''b'' for given sides ''a'' and ''b''. Menaechmus (circa 350 BC) considered the problem geometrically by intersecting the pair of plane conics ''ay'' = ''x''2 and ''xy'' = ''ab''. In the 3rd century BC, Archimedes and Apollonius of Perga, Apollonius systematically studied additional problems on conic sections using coordinates. Mathematics in medieval Islam, Medieval Muslim mathematicians, including Ibn al-Haytham in the 10th century AD, solved certain cubic equations by purely algebraic means and then interpreted the results geometrically. The Persian people, Persian mathematician Omar Khayyám (born 1048 AD) discovered a method for solving cubic equations by intersecting a parabola with a circle and seems to have been the first to conceive a general theory of cubic equations. A few years after Omar Khayyám, Sharaf al-Din al-Tusi's ''Treatise on equations'' has been described as "inaugurating the beginning of algebraic geometry".

## Renaissance

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, 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. 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, Newton and Gottfried Wilhelm Leibniz, 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, Lagrange and Leonhard Euler, Euler.

## 19th and early 20th century

It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals 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 forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other types 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 transformations. This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism. The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces. In the same period began the algebraization of the algebraic geometry through
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
. The prominent results in this direction are
Hilbert's basis theorem In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. Statement If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. David H ...
and
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz ' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry.

## 20th century

B. L. van der Waerden, Oscar Zariski and André Weil developed a foundation for algebraic geometry based on contemporary
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, including valuation theory and the theory of ideal (ring theory), ideals. One of the goals was to give a rigorous framework for proving the results of Italian school of algebraic geometry. In particular, this school used systematically the notion of generic point without any precise definition, which was first given by these authors during the 1930s. In the 1950s and 1960s, Jean-Pierre Serre and
Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

recast the foundations making use of
sheaf theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Later, from about 1960, and largely led by Grothendieck, 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 stabilized in the 1970s, and new applications were made, both to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

and to more classical geometric questions on algebraic varieties, singularity theory, singularities, moduli space, moduli, and formal 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 In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
s, which have a rich theory. They were instrumental in the proof of
Fermat's last theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
and are also used in elliptic-curve cryptography. In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory of Gröbner bases, introduced by
Bruno Buchberger Bruno Buchberger (born 22 October 1942) is Professor of Computer Mathematics at Johannes Kepler University in Linz Linz (; ; cs, Linec) is the capital of Upper Austria and List of cities and towns in Austria, third-largest city in Austria. In ...

in 1965. Another founding method, more specially devoted to real algebraic geometry, is the cylindrical algebraic decomposition, introduced by
George E. CollinsGeorge E. Collins (January 10, 1928 in Stuart, Iowa Stuart is a city in Lincoln Township, Adair County, and in Stuart Township, Guthrie County, Iowa, Guthrie County, in the U.S. state of Iowa. That part of the city within Guthrie County is part o ...

# Analytic geometry

An analytic variety is defined locally as the set of common solutions of several equations involving analytic functions. It is analogous to the included concept of real or complex algebraic variety. Any complex manifold is an analytic variety. Since analytic varieties may have Mathematical singularity, singular points, not all analytic varieties are manifolds. Modern analytic geometry is essentially equivalent to real and complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper ''GAGA'', the name of which is French for ''Algebraic geometry and analytic geometry''. Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite Characteristic (algebra), characteristic.

# Applications

Algebraic geometry now finds applications in algebraic statistics, statistics, control theory, robotics, algebraic geometric code, error-correcting codes, computational phylogenetics, phylogenetics and geometric modelling. There are also connections to Homological mirror symmetry, string theory, game theory, Matching (graph theory), graph matchings, solitons and integer programming.

* Algebraic statistics * Differential geometry * Complex geometry * Geometric algebra * Glossary of classical algebraic geometry * Intersection theory * List of publications in mathematics#Algebraic geometry, Important publications in algebraic geometry * List of algebraic surfaces * Noncommutative algebraic geometry * Diffiety, Diffiety theory * Differential algebraic geometry *
Real algebraic geometryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
* Nonlinear algebra *Geometrically (algebraic geometry)

# References

## Sources

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;Some classic textbooks that predate schemes: * * * * ;Modern textbooks that do not use the language of schemes: * * * * * * ;Textbooks in computational algebraic geometry * * * * * * * ;Textbooks and references for schemes: * * * * * *