π is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of π. that is, π may be an isomorphism on "e". this happens, for example, in the trivial situation where formula_44 is already an invertible sheaf. in particular, in such cases the morphism π does not determine the exceptional divisor. another situation where the exceptional locus can be strictly smaller than the exceptional divisor is when "x" has singularities. for instance, let "x" be the affine cone over . "x" can be given as the vanishing locus of in a. the ideals and define two planes, each of which passes through the vertex of "x". away from the vertex, these planes are hypersurfaces in "x", so the blowup is an isomorphism there. the exceptional locus of the blowup of either of these planes is therefore centered over the vertex of the cone, and consequently it is strictly smaller than the exceptional divisor.