quasiprojective variety (idea)

This is a basic description of projective closed sets and quasiprojective varieties, a slight generalisation of affine spaces. Slight differences in terminology exist: I've used that of Shafarevich's Basic Algebraic Geometry.

I'll use the ideas that Noether explained in Zariski topology and projective space, but with slight changes of terminology and notation: Noether's affine algebraic varieties will become affine closed sets, and the coordinates of a point in projective space will be written (a₀:a₁:...:a_n).

Our goals here are (1) to find something analagous to affine closed sets in projective space, and (2) to find a phenomenon of which affine and projective closed sets are special cases.

Owing to the way that projective space is described, the zero set of a polynomial is not well defined: in P¹, for example, what is the zero set of f(x) = x₀ + 1? At first glance, (-1:0) should be in it since -1 + 1 = 0. However, in projective space (-1:0) is the same point as (1:0), and this isn't in the zero set. So the zero set of the polynomial on projective space is not well defined.

The solution is to allow only homogeneous polynomials: thus if f(x) = 0 when x is written as (x₀:...:x_n), if we multiply all the coordinates by a constant, a, to get another form for x, then f(x) = a^d*0 = 0, where d is the degree of f. Thus we find that we have a well defined concept: the zero set in Pⁿ of a (finite) collection of homogeneous polynomials, and we call this a projective closed set.

The open sets corresponding to these closed sets obey the axioms for a topology the same way as affine closed set. We now need to find how the two concepts connect.

As Noether explained, a projective space Pⁿ can be thought of as the vector space kⁿ together with a copy of P^n-1at infinity. However, it can also be though of as n+1 copies of kⁿ overlapping in such a way that the points at infinity are covered. Consider, for instance, the complex P¹, aka the Riemann sphere. By removing the point at infinity (the north pole), this can be flattened back down to recover your original complex plane. Alternatively, though, you could remove the south pole, zero, and flatten it another way, leaving a new complex plane with your original infinity as the origin.

In general, this is done by picking one of the coordinates x_i to be nonzero, and setting it to 1 by dividing through. You then have one coordinate that is fixed, and n remaining ones that you can choose freely from the field k: in other words, you have something isomorphic to kⁿ, which we call A_iⁿ. We find that all of our original affine closed sets can be produced as the intersection of A_iⁿ with closed projective sets. Unfortunately, in our new topology on Pⁿ, these sets turn out to be open. Thus we define a new concept, the simplest construction of which affine and projective closed sets are special cases: the quasiprojective variety.

A quasiprojective variety is defined as an open subset of a projective closed set. Projective closed sets are clearly quasiprojective varieties, since every set in a topology is open as a subset of itself. To show that an affine closed set, A, is a quasiprojective variety, recall that it is given by the intersection of a closed projective set, X, with an A_iⁿ. The complement of A in X is given by the equation x_i = 0 and so is closed as a subset of X. Thus A is open in X.

Projective and quasiprojective varieties have many interesting properties and useful applications. But they may have to wait for another node.