Editing Algebraic geometry (section)

=== 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.