In mathematics, blowing up or blowup is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. At a point that is being 'blown up' (the metaphor is inflation of a balloon, rather than explosion), is replaced by the whole space of tangent directions at (which, more formally, can be defined as the projective space constructed from the tangent space at ). More general blow-ups are also defined.
Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. It may also be considered from an extrinsic point of view; for example by taking a plane curve and applying a transformation to the projective plane in which it sits. This is in fact the more classical approach, and this is reflected in some of the terminology. Blowing up is also more formally a monoidal transformation; in the projective plane simply blowing up one point takes one to a quadric, and a curve must be blown down to return to the plane. That is, transformations in the Cremona group are not 'monoidal' or single-centred. See also quadratic transformation.
Let be the origin in -dimensional complex space, . That is, is the point where the coordinate functions simultaneously vanish. Let be -dimensional complex projective space with homogeneous coordinates . Let be the subset of that satisfies simultaneously the equations for . The projection
naturally induces a holomorphic map
This map (or, often, the space ) is called the blow-up (variously spelled blow up or blowup) of .
The exceptional divisor is defined as the inverse image of the blow-up locus under . It is easy to see that
is a copy of projective space. It is an effective divisor. Away from , is an isomorphism between and ; it is a birational map between and .
More generally, one can blow up any codimension- complex submanifold of . Suppose that is the locus of the equations , and let be homogeneous coordinates on . Then the blow-up is the locus of the equations for all and , in the space .
More generally still, one can blow up any submanifold of any complex manifold by applying this construction locally. The effect is, as before, to replace the blow-up locus with the exceptional divisor . In other words, the blow-up map
Since is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; is the only smooth complex representative of its homology class in . (Suppose could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of .) This is why the divisor is called exceptional.
Let be some submanifold of other than . If is disjoint from , then it is essentially unaffected by blowing up along . However, if it intersects , then there are two distinct analogues of in the blow-up . One is the proper (or strict) transform, which is the closure of ; its normal bundle in is typically different from that of in . The other is the total transform, which incorporates some or all of ; it is essentially the pullback of in cohomology.
In the blow-up of 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 at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere 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 and a closed subscheme , one blows up in (or ). Then
is a fibration. The general fiber is naturally isomorphic to , while the central fiber is a union of two schemes: one is the blow-up of along , and the other is the normal cone of with its fibers completed to projective spaces.
Blow-ups can also be performed in the symplectic category, by endowing the symplectic manifold with a compatible almost complex structure and proceeding with a complex blow-up. This makes sense on a purely topological level; however, endowing the blow-up with a symplectic form requires some care, because one cannot arbitrarily extend the symplectic form across the exceptional divisor . One must alter the symplectic form in a neighborhood of , or perform the blow-up by cutting out a neighborhood of and collapsing the boundary in a well-defined way. This is best understood using the formalism of symplectic cutting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of symplectic summation, is the symplectic analogue of deformation to the normal cone along a smooth divisor.