An affine variety V is a variety contained in affine space. For example,
is the cone, and
is a conic section, which is a subvariety of the cone. The cone can be written
to indicate that it is the variety corresponding to 
Naturally, many other polynomials vanish on , in fact all polynomials in 
.The set I(C) is an ideal in the polynomial ring 
. Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by 
and 
.
A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map
is a morphism from 
to 
. Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety 
is isomorphic to the cone via the coordinate change 
.
Many polynomials f may be factored, for instance
,and then 
. Consequently, only [u]irreducible polynomials ,and more generally only prime ideals 
are used in the definition of a variety. An affine variety V is the set of common zeros of a collection of polynomials 
, ..., 
, i.e.,
as long as the ideal
is a prime ideal. More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an algebraic set. Most points in V will have dimension 
, but V may have singular points like the origin in the cone.
When V is one-dimensional generically (at almost all points), which typically occurs when
, then V is called a curve. When V is two-dimensional, it is called a surface. In the case of complex affine space, a curve is a Riemann surface, possibly with some singularities.
Mathematica has a built-in function ImplicitPlot in the Mathematica add-on package Graphics`ImplicitPlot` (which can be loaded with the command <<Graphics`) that will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle.
<<Graphics`;
Show[GraphicsArray[{
ImplicitPlot[x^2 - y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity],
ImplicitPlot[x^2 + y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity]
}]]
An extension to this function called ImplicitPlot3D (Wilkinson) that can be used to plot affine varieties in three-dimensional space can be downloaded from the Mathematica Information Center.
(1)
is the cone, and
(2)
is a conic section, which is a subvariety of the cone. The cone can be written
to indicate that it is the variety corresponding to Naturally, many other polynomials vanish on , in fact all polynomials in .The set I(C) is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map
is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .Many polynomials f may be factored, for instance
,and then . Consequently, only [u]irreducible polynomials ,and more generally only prime ideals are used in the definition of a variety. An affine variety V is the set of common zeros of a collection of polynomials , ..., , i.e., (3)
as long as the ideal
is a prime ideal. More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an algebraic set. Most points in V will have dimension , but V may have singular points like the origin in the cone.When V is one-dimensional generically (at almost all points), which typically occurs when
, then V is called a curve. When V is two-dimensional, it is called a surface. In the case of complex affine space, a curve is a Riemann surface, possibly with some singularities.Mathematica has a built-in function ImplicitPlot in the Mathematica add-on package Graphics`ImplicitPlot` (which can be loaded with the command <<Graphics`) that will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle.
<<Graphics`;
Show[GraphicsArray[{
ImplicitPlot[x^2 - y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity],
ImplicitPlot[x^2 + y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity]
}]]
An extension to this function called ImplicitPlot3D (Wilkinson) that can be used to plot affine varieties in three-dimensional space can be downloaded from the Mathematica Information Center.
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