Blowing Up - The Blowup of A Point in A Plane

The Blowup of A Point in A Plane

The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example.

The blowup has a synthetic description as an incidence correspondence. Recall that the Grassmannian G(1,2) parameterizes the set of all lines in the projective plane. The blowup of the projective plane at the point P, which we will denote X, is

X is a projective variety because it is a closed subvariety of a product of projective varieties. It comes with a natural morphism π to P2 that takes the pair to Q. This morphism is an isomorphism on the open subset of all points with Q ≠ P because the line is determined by those two points. When Q = P, however, the line can be any line through P. These lines correspond to the space of directions through P, which is isomorphic to P1. This P1 is called the exceptional divisor, and by definition it is the projectivized normal space at P. Because P is a point, the normal space is the same as the tangent space, so the exceptional divisor is isomorphic to the projectivized tangent space at P.

To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P2 homogeneous coordinates in which P is the point . By projective duality, G(1,2) is isomorphic to P2, so we may give it homogenous coordinates . A line is the set of all such that X₀L₀ + X₁L₁ + X₂L₂ = 0. Therefore, the blowup can be described as

The blowup is an isomorphism away from P, and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation, we may assume that P = . Write x and y for the coordinates on the affine plane X₂≠0. The condition P ∈ implies that L₂ = 0, so we may replace the Grassmannian with a P1. Then the blowup is the variety

It is more common to change coordinates so as to reverse one of the signs. Then the blowup can be written as

This equation is easier to generalize than the previous one.

The blowup can also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane A2. The normal space to the origin is the vector space m/m2, where m = (x, y) is the maximal ideal of the origin. Algebraically, the projectivization of this vector space is Proj of its symmetric algebra, that is,

In this example, this has a concrete description as

where x and y have degree 0 and z and w have degree 1.

Over the real or complex numbers, the blowup has a topological description as the connected sum . Assume that P is the origin in A2 ⊆ P2, and write L for the line at infinity. A2 \ {0} has an inversion map t which sends (x, y) to (x/(|x|2 + |y|2), y/(|x|2 + |y|2)). t is the circle inversion with respect to the unit sphere S: It fixes S, preserves each line through the origin, and exchanges the inside of the sphere with the outside. t extends to a continuous map P2 → A2 by sending the line at infinity to the origin. This extension, which we also denote t, can be used to construct the blowup. Let C denote the complement of the unit ball. The blowup X is the manifold obtained by attaching two copies of C along S. X comes with a map π to P2 which is the identity on the first copy of C and t on the second copy of C. This map is an isomorphism away from P, and the fiber over P is the line at infinity in the second copy of C. Each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin.

For CP2 this process ought to produce an oriented manifold. In order to make this happen, the two copies of C should be given opposite orientations. In symbols, X is, where is CP2 with the opposite of the standard orientation.