Editing Algebraic geometry (section)

==Basic notions==
{{Further|Algebraic variety}}

=== Zeros of simultaneous polynomials ===
[[File:Slanted circle.png|thumb|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 [[systems of polynomial equations|polynomial equations]]. For instance, the [[N-sphere|two-dimensional]] [[sphere]] of radius 1 in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup> could be defined as the set of all points <math>(x, y, z)</math> with

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

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

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

=== Affine varieties ===
{{main|Affine variety}}

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 consider the [[affine space]] of dimension ''n'' over ''k'', denoted '''A'''<sup>n</sup>(''k'') (or more simply '''A'''<sup>''n''</sup>, when ''k'' is clear from the context). When one fixes a coordinate system, one may identify '''A'''<sup>n</sup>(''k'') with ''k''<sup>''n''</sup>. The purpose of not working with ''k''<sup>''n''</sup> is to emphasize that one "forgets" the vector space structure that ''k''<sup>n</sup> carries.

A function ''f'' : '''A'''<sup>''n''</sup> → '''A'''<sup>1</sup> 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''<sub>1</sub>,...,''x''<sub>''n''</sub>] such that ''f''(''M'') = ''p''(''t''<sub>1</sub>,...,''t''<sub>''n''</sub>) for every point ''M'' with coordinates (''t''<sub>1</sub>,...,''t''<sub>''n''</sub>) in '''A'''<sup>''n''</sup>. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in '''A'''<sup>''n''</sup>.

When a coordinate system is chosen, the regular functions on the affine ''n''-space may be identified with the ring of [[polynomial function]]s in ''n'' variables over ''k''. Therefore, the set of the regular functions on '''A'''<sup>''n''</sup> is a ring, which is denoted ''k''['''A'''<sup>''n''</sup>].

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'''<sup>n</sup>]. The ''vanishing set of S'' (or ''vanishing locus'' or ''zero set'') is the set ''V''(''S'') of all points in '''A'''<sup>''n''</sup> where every polynomial in ''S'' vanishes. Symbolically,

:<math>V(S) = \{(t_1,\dots,t_n) \mid p(t_1,\dots,t_n) = 0 \text{ for all } p \in S\}.\,</math>

A subset of '''A'''<sup>''n''</sup> 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'''<sup>''n''</sup>, can one recover the set of polynomials which generate it? If ''U'' is ''any'' subset of '''A'''<sup>''n''</sup>, 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 the polynomial ring ''k''['''A'''<sup>''n''</sup>].

Two natural questions to ask are:
* Given a subset ''U'' of '''A'''<sup>''n''</sup>, 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'''<sup>''n''</sup> whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of ''k''['''A'''<sup>''n''</sup>]. 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]]. In one of its forms, it says that ''I''(''V''(''S'')) is the [[radical of an ideal|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'''<sup>''n''</sup>] 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. 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 ''[[algebraic variety|variety]]''. 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]] 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 ===
{{main|Regular function}}

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'' or ''polynomial functions''. A regular function on an algebraic set ''V'' contained in '''A'''<sup>''n''</sup> is the restriction to ''V'' of a regular function on '''A'''<sup>''n''</sup>. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even [[analytic function|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 [[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'''<sup>''n''</sup>, 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'''<sup>''n''</sup>], then ''f''&nbsp;&minus;&nbsp;''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'''<sup>''n''</sup>]/''I''(''V'').

=== Morphism of affine varieties ===
Using regular functions from an affine variety to '''A'''<sup>1</sup>, we can define [[morphism of algebraic varieties|regular map]]s 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'''<sup>''n''</sup>. Choose ''m'' regular functions on ''V'', and call them ''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>. We define a ''regular map'' ''f'' from ''V'' to '''A'''<sup>''m''</sup> by letting {{nowrap|1=''f'' = (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)}}. In other words, each ''f''<sub>''i''</sub> determines one coordinate of the [[image (mathematics)|range]] of ''f''.

If ''V''′ is a variety contained in '''A'''<sup>''m''</sup>, 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 theory|category]], where the objects are the affine algebraic sets and the [[morphism]]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 {{nowrap|''f'' ∘ ''g'' ∈ ''k''[''V'']}}. The map {{nowrap|''f'' → ''f'' ∘ ''g''}} is a [[ring homomorphism]] 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]] between the category of algebraic sets and the [[dual (category theory)|opposite category]] of the finitely generated [[reduced ring|reduced]] ''k''-algebras. This equivalence is one of the starting points of [[scheme theory]].

=== Rational function and birational equivalence ===
{{main|Rational mapping}}
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]] and has thus a [[field of fractions]] which is denoted ''k''(''V'') and called the ''field of the rational functions'' on ''V'' or, shortly, the ''[[function field of an algebraic variety|function field]]'' of ''V''. Its elements are the restrictions to ''V'' of the [[rational function]]s over the affine space containing ''V''. The [[domain of a function|domain]] of a rational function ''f'' is not ''V'' but the [[complement (set theory)|complement]] 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''<nowiki>'</nowiki>. As with the regular maps, the rational maps from ''V'' to ''V''<nowiki>'</nowiki> may be identified to the [[ring homomorphism|field homomorphism]]s from ''k''(''V''<nowiki>'</nowiki>) to ''k''(''V'').

Two affine varieties are ''birationally equivalent'' if there are two rational functions between them which are [[function inverse|inverse]] 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 variety]]'' if it is birationally equivalent to an affine space. This means that the variety admits a ''rational parameterization'', that is a [[Parametrization (geometry)|parametrization]] with [[rational function]]s. For example, the circle of equation <math>x^2+y^2-1=0</math> is a rational curve, as it has the [[parametric equation]]
:<math>x=\frac{2\,t}{1+t^2}</math>
:<math>y=\frac{1-t^2}{1+t^2}\,,</math>
which may also be viewed as a rational map from the line to the circle.

The problem of [[resolution of singularities]] 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 (algebra)|characteristic]] 0 by [[Heisuke Hironaka]] in 1964 and is yet unsolved in finite characteristic.

=== Projective variety ===
{{Main|Algebraic geometry of projective spaces}}

[[File:Parabola & cubic curve in projective space.png|thumb|Parabola ({{nowrap|1=''y'' = ''x''<sup>2</sup>}}, red) and cubic ({{nowrap|1=''y'' = ''x''<sup>3</sup>}}, blue) in projective space]]
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 {{nowrap|''x''<sup>2</sup> + 1}}, 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 {{nowrap|''V''(''y'' &minus; ''x''<sup>2</sup>)}}. If we draw it, we get a [[parabola]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'',&nbsp;''x''<sup>2</sup>) 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''&nbsp;&minus;&nbsp;''x''<sup>3</sup>). This is a [[cubic curve]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'',&nbsp;''x''<sup>3</sup>) 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''&nbsp;&minus;&nbsp;''x''<sup>3</sup>) is different from the behavior "at infinity" of ''V''(''y''&nbsp;&minus;&nbsp;''x''<sup>2</sup>).

The consideration of the ''projective completion'' of the two curves, which is their prolongation "at infinity" in the [[projective plane]], allows us to quantify this difference: the point at infinity of the parabola is a [[regular point of an algebraic variety|regular point]], whose tangent is the [[line at infinity]], while the point at infinity of the cubic curve is a [[cusp (singularity)|cusp]]. Also, both curves are rational, as they are parameterized by ''x'', and the [[Riemann-Roch theorem for algebraic curves|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]] 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]]'' '''P'''<sup>''n''</sup> 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 {{nowrap|''n'' + 1}}, or equivalently to the set of the vector lines in a vector space of dimension {{nowrap|''n'' + 1}}. When a coordinate system has been chosen in the space of dimension {{nowrap|''n'' + 1}}, 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]] of a point of '''P'''<sup>''n''</sup> as a sequence of {{nowrap|''n'' + 1}} 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 {{nowrap|''n'' + 1}} variables vanishes at all points of a line passing through the origin if and only if it is [[Homogeneous polynomial|homogeneous]]. In this case, one says that the polynomial ''vanishes'' at the corresponding point of '''P'''<sup>''n''</sup>. This allows us to define a ''projective algebraic set'' in '''P'''<sup>''n''</sup> as the set {{nowrap|''V''(''f''<sub>1</sub>, ..., ''f''<sub>''k''</sub>)}}, where a finite set of homogeneous polynomials {{nowrap|{''f''<sub>1</sub>, ..., ''f''<sub>''k''</sub>} }} vanishes. Like for affine algebraic sets, there is a [[bijection]] between the projective algebraic sets and the reduced [[homogeneous ideal]]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]] is an [[integral domain]], the ''projective coordinates ring'' being defined as the quotient of the graded ring or the polynomials in {{nowrap|''n'' + 1}} 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.