here "q" denotes another point and formula_2 is an element of the grassmannian. "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 p that takes the pair formula_3 to "q". this morphism is an isomorphism on the open subset of all points formula_3 with "q" ≠ "p" because the line formula_2 is determined by those two points. when "q" = "p", however, the line formula_2 can be any line through "p". these lines correspond to the space of directions through "p", which is isomorphic to p. this p 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".