along with a morphism such that is an invertible sheaf, characterized by this universal property: for any morphism f: Y → X such that is an invertible sheaf, f factors uniquely through π. Notice that has this property; this is how the blow-up is constructed. Here Proj is the Proj construction on graded sheaves of commutative rings. Exceptional divisors The exceptional divisor of a blowup is the subscheme defined by the inverse image of the ideal sheaf , which is sometimes denoted . It follows from the definition of the blow up in terms of Proj that this subscheme E is defined by the ideal sheaf . This ideal sheaf is also the relative for π. 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 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 A4. 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. Further examples Blowups of linear subspaces Let be -dimensional projective space. Fix a linear subspace of codimension . There are several explicit ways to describe the blowup of along . Suppose that has coordinates . After changing coordinates, we may assume that . The blowup may be embedded in . Let be coordinates on the second factor. Because is defined by a regular sequence, the blowup is determined by the vanishing of the two-by-two minors of the matrix This system of equations is equivalent to asserting that the two rows are linearly dependent. A point is in if and only if, when its coordinates are substituted in the first row of the matrix above, that row is zero. In this case, there are