Infinity is quite the tricky concept in mathematics. While there’s a basic concept of what “infinity” means in the foundations of mathematics, even non-mathematicians will probably recognize that it’s a very challenging concept to grapple with. Still, some notion related to infinity might be convenient to have in many branches of mathematics, so these different branches will usually come up with some “variant” of the concept of infinity that will be easier to deal with yet still provide the convenience we desire from such a concept.
In this post we discuss one way in which we incorporate the concept of infinity in algebraic geometry. We often hear, for example, of how parallel lines “meet at infinity”; it is the goal of this post to somehow give the idea of what that statement means. In addition, this concept can be convenient for many purposes, for instance in the study of elliptic curves.
Consider lines in the – plane, with the additional condition that they must pass through the origin. The equations of these lines are always of the form
since these lines must have zero -intercept or -intercept in order for them to pass through the origin. A horizontal line corresponds to the equation
i.e. , while a diagonal line has
i.e. . Meanwhile, a vertical line corresponds to the equation
which one may express, alternatively, as ““.
Let us now consider another line, a vertical line at , and look at where our lines which pass through the origin meet this vertical line. The horizontal line meets the vertical line at the point . The diagonal line meets the vertical line at the point . The vertical line does not meet the vertical line at all. More generally, every line that passes through the origin, except the vertical line , meets the vertical line at a single point. Conversely, at every point on the vertical line there is one line that passes through the origin and also passes through that point on the vertical line .
So the set of lines on the plane that pass through the origin and the set of points on the line are almost in one-to-one correspondence with one another, except that there is one line, the vertical line , which does not correspond to any point on the line.
This gives us a method of obtaining a version of the (real) line with a “point at infinity”. We declare the “points” of this “line” to be the lines in the plane that pass through the origin, and the vertical line that passes through the origin is declared to be the “point at infinity”. This “line” is called the (real) projective line. Note that it is not really a line with a set of points, but a set of lines in the plane that pass through the origin.
The procedure we performed earlier, setting up a vertical line at and looking at the points where it is met by the lines passing through the origin, is analogous to looking at the one-dimensional “shadows” or “projections” of these lines, and gives us the reasoning behind the term “projective line”. The “point at infinity” is that one line that has no shadow.
We can generalize this idea to form the (real) projective plane. The “points” of this “plane” are lines in three-dimensional space that pass through the origin. The “lines” of this “plane” are planes that pass that pass through the origin. Note that two parallel lines are merely “shadows” or “projections” of two planes that pass through the origin, and their intersection is a line through the origin, i.e. a point in the projective plane, which happens to be a “point at infinity”. If we set up another plane such that the two parallel lines show up as the “shadows” of planes, then the line where the two planes intersect will have no “shadow”. This is what we mean when we say that two parallel lines intersect at infinity.
We can actually generalize this to any dimension ; the set of lines through the origin in this -dimensional space form what we call (real) projective space, written .
We can also generalize this construction to the complex numbers; we simply take the set of all lines where , , and are all complex numbers. Since the complex numbers form a plane, adding a “point at infinity” makes the complex projective line into a sphere. One can perhaps imagine this as “rolling up” the plane into a sphere, and the “point at infinity” “closes” the sphere. A more formal method of visualizing this relation is provided by the technique called “stereographic projection“, which is used by mapmakers to relate a globe to a flat map.
The complex projective line, written has a special name; it is called the Riemann sphere. We also have higher-dimensional generalizations, called complex projective space, and written .
There are other ways of defining projective spaces. For instance, the real projective line can also be defined as the set of points of a circle centered at the origin with “antipodal” points “identified” via an equivalence relation (see Modular Arithmetic and Quotient Sets). Since a line in the plane through the origin crosses a circle at two points which are “antipodal” (“opposite”, in some sense), this is equivalent to our earlier definition in terms of lines through the origin. For higher-dimensional generalizations, we simply replace the circle by higher-dimensional spheres centered at the origin.
We mention one more method of defining projective spaces which is useful in algebraic geometry (see Basics of Algebraic Geometry). We define the n-dimensional projective space as the set of equivalence classes of points specified by (real or complex) coordinates under the equivalence relation
where is any nonzero constant. The idea is that the points and for all nonzero constants all lie on a single line which happens to pass through the origin.
In order to do algebraic geometry on projective space, the polynomials being studied need to be homogeneous, which means that all the terms of the polynomial must have the same degree. For instance, we cannot have
in projective space; instead we can only use something like
which is homogeneous. This is necessary to protect the defining property
of our projective space. However, there is a procedure for obtaining the ordinary (called “affine“) space from a projective space, similar to the “projection” construction described above, and in this case the homogeneous polynomial in projective space becomes in affine space.
Algebraic Geometry by Robin Hartshorne