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 [X0:X1:X2] in which P is the point [P0:P1:P2]. By projective duality, G(1,2) is isomorphic to P2, so we may give it homogeneous coordinates [L0:L1:L2]. A line is the set of all [X0:X1:X2] 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 = [0:0:1]. 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 be easily visualized if we remove the infinity point of the Grassmannian, e.g. by setting w = 1, and obtain the standard saddle surface y = xz in 3D space. 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