no conditions on . If, however, that row is non-zero, then linear dependence implies that the second row is a scalar multiple of the first and therefore that there is a unique point such that is in the blowup. This blowup can also be given a synthetic description as the incidence correspondence where denotes the Grassmannian of -dimensional subspaces in . To see the relation with the previous coordinatization, observe that the set of all that contain is isomorphic to a projective space . This is because each subspace is the linear join of and a point not in , and two points and determine the same if and only if they have the same image under the projection of away from . Therefore, the Grassmannian may be replaced by a copy of . When , there is only one subspace containing , the linear join of and . In the coordinates above, this is the case where is not the zero vector. The case corresponds to being the zero vector, and in this case, any is allowed, that is, any containing is possible. Blowing up intersections of curves scheme-theoretically Let be generic homogeneous polynomials of degree (meaning their associated projective varieties intersects at points by Bézout's theorem). The following projective morphism of schemes gives a model of blowing up at points: Looking at the fibers explains why this is true: if we take a point then the pullback diagram tells us the fiber is a point whenever or and the fiber is if . Related constructions In the blow-up of Cn described above, there was nothing essential about the use of complex numbers; blow-ups can be performed over any field. For example, the real blow-up of R2 at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere S2 results in the real projective plane. Deformation to the normal cone is a blow-up technique used to prove many results in algebraic geometry. Given a scheme X and a closed subscheme V, one blows up Then is a fibration. The general fiber is naturally isomorphic to X, while the central fiber is a union of two schemes: one is the blow-up of X along V, and the other is the normal cone of V with its fibers completed to projective spaces. Blow-ups can also be performed in the symplectic category, by endowing the symplectic manifold with a