Projective Varieties, An Introduction

Projective Space

Real space, or euclidean space, is described by real coordinates running along perpendicular axes. For example, 3 space, denoted R³, is the 3 dimensional world we live in, with x y and z coordinates. In contrast, projective space consists of lines passing through the origin, such as the x axis, or 2x = 5y = 3z. These lines become the "points" of projective space.

It is sometimes convenient to intersect these lines with the unit sphere. Thus the points of the sphere become the points of projective space - but there is a catch. Each line intersects the sphere in two points, the ends of a diameter. Thus the points of the sphere define projective space, as long as you remember that each point is essentially the same as the point on the other side. Travel halfway around the sphere, and you are back to start.

The sphere is a 2 dimensional shape, living in a 3 dimensional world. Thus the 2-sphere, living in R³, is denoted S². In the same way, projective space is considered 2 dimensional. Therefore, the lines passing through the origin in R³ build a projective space that is denoted P².

This can be generalized to an arbitrary field K. Normal space is defined by n coordinates, and is denoted Kⁿ. Projective space consists of lines in Kⁿ that pass through the origin.

Homogeneous Polynomial

A polynomial p, in x₁ through x_n, with coefficients in K, defines a function from Kⁿ into K. The region R is the set of points in Kⁿ that vanishes on p. But what about projective space? If p is homogeneous, i.e. each term has degree d, We can scale the input variables x₁ through x_n by a factor l, and p(x₁…x_n) is multiplied by l^d. The value of the polynomial is zero, or nonzero, regardless of the scaling factor l. Therefore, p vanishes on certain lines passing through the origin. In other words, p defines a region R in projective space where p(R) = 0.

Conversely, suppose p is not homogeneous. Group terms together by degree, so that the first block of terms has degree d₁, the next block has degree d₂, and so on. Let these blocks define homogeneous polynomials q₁ q₂ etc. Without dwelling on the subject, one can always find a point b in Kⁿ (or at least its closure) that leaves all of these subpolynomials, i.e. their product, nonzero. (This is really weak nullstellensatz.) Let q_i have the value v_i at b, where some v_i is nonzero. Scale the inputs by l, which multiplies each v_i by l to the d_i. This defines a polynomial in l, with coefficients v_i. Such a polynomial has finitely many roots. There are points along this line, passing through b and the origin, where p is zero, and where p is nonzero. It is not clear how to define R in projective space such that p vanishes on R.

In summary, the polynomials that vanish on specific regions in projective space are the homogeneous polynomials, or ideals built from these.

Graded Ring

The ring of polynomials over a field K is an example of a graded ring. A homogeneous ideal H in this ring is generated by homogeneous polynomials, and it has the happy property that any polynomial in H can be separated into homogeneous blocks, as described above, and these homogeneous subpolynomials are part of H. If these concepts are foreign to you, you should stop and review the properties of graded rings, and their homogeneous ideals.