Orbit-spaces

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Algebraic Geometry for Non-Experts

Orbit Spaces

Now we shall introduce the variety, which is a geometric object. This can be a curve (as in our case), a plane, or a point. We then proceed to define an orbit, which is a subset of a variety. An example will show how we can associate an orbit in a plane with a point in another plane, which shall be our orbit space.

Definition of a variety

We are familiar with the concept of a curve in the xy-plane given by the solutions to a polynomial equation. There are, however, equations that define geometric objects that are not curves. For example, the solution to (x-1)(y-2) = 0 is the single point (1,2). Another example is the solution to the 0-polynomial, which is the whole plane, the 0-polynomial being zero for all x and y. The point and the plane are certainly not curves. We need a name for the geometric object that can be a point, a curve, or a plane, all defined by polynomial equations. This geometric object shall be called a variety.

Remark. It is interesting to note that to be able to be even more general, we introduce in algebraic geometry an object called a scheme. As a curve is an example of a variety, a variety is an example of a scheme. The language of algebraic geometry is built on the theory of schemes. For example, two parallel lines can be described by a single scheme.

Definition. A variety is the geometric object that represents solutions to one or more polynomial equations.

It is not strictly necessary to define the variety to explain the ideas in this note. The idea of a variety is so essential, however, in algebraic geometry that it has been defined. If you feel uncomfortable about calling something a variety, call it a curve and accept that a curve can be a point or the whole plane.

Definition of an orbit

Assume we have a variety that is the whole xy-plane. The two points (1,2) and (2,1) is an example of an orbit in this variety. It is called an orbit because it is that it is possible to go through all points in the orbit by performing a transformation. The transformation in the above example takes x to y and y to x so that (x,y) becomes (y,x). In the example, orbit (1,2) becomes (2,1) and visa versa. The points in these orbits lie as mirror images of the line x = y in the xy-plane. See the two points in the coordinate system in the left of Figure 1. This specific transformation belongs to a group of transformations called the permutation groups, that, as the name indicates, permutes the coordinates.

The orbit space

We now have a variety, the xy-plane, which is divided into orbits such as {(a,b),(b,a)}. We shall, by example, show that there exists another plane, the st-plane, which has points that are in a one-to-one correspondence with the orbits in the xy-plane. This means that every point of the st-plane corresponds to an orbit in the xy-plane. This st-plane is called an orbit space. It can be shown that the relationship between the xy-coordinates and st-coordinates is s = x + y and t = xy.

Figure 1: Each point in the st-plane corresponds to an orbit in the xy-plane.

We first pick a point in the st-plane and then calculate the corresponding points in the xy-plane. We than prove that the two points are in the same orbit.

Remember that s = x + y and t = xy. Substituting y = t/x into s = x + y gives s = x + t/x, which is the same as

x2 - xs + t = 0.

Now, given a point (s,t) in the st-plane, we can by the above equation find which points it corresponds to in the xy-plane. Generally, the x-values will be the roots of x2 - sx + t = 0 and the y-values are given by y = s - x. In the st-plane in the figure the example point is (1.5,0.5). This gives us a second order equation, x2 - 1.5x + 0.5 = 0, which has roots x1 = 1.5/2 + sqrt(1.52 - 4*0.5)/2 = 1 and x2 = 1.5/2 - sqrt(1.52 - 4*0.5)/2 = 0.5.

The y-values are y1 = s - x1 = 1.5 - 1 = 0.5 and y2 = s - x2 = 1.5 - 0.5 = 1.

Now (x1,y1) = (1,0.5) and (x2,y2) = (0.5,1) are clearly contained in an orbit in the xy-plane.

Remark. We will only get real x-values if s2 - 4t > 0. We should therefore also consider orbits in the complex plane. But as usual, drawing is only possible in the real plane.

Conclusion

We have shown that each orbit of the xy-plane, can be associated with points in the st-plane when the orbits are on the form {(a,b),(b,a)}. This st-plane was called the orbit space, which, again, is a central object of study in invariant theory and deformation theory, where one studies how to pass from one orbit space to another.