Blowing up

The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion.

A consequence of this is that blowups can be used to resolve the singularities of birational maps.

This is reflected in some of the terminology, such as the classical term monoidal transformation.

From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor.

is called the exceptional divisor, and by definition it is the projectivized normal space at

To give coordinates on the blowup, we can write down equations for the above incidence correspondence.

Then the blowup is the variety It is more common to change coordinates so as to reverse one of the signs.

Then the blowup can be written as This equation is easier to generalize than the previous one.

The blowup can be easily visualized if we remove the infinity point of the Grassmannian, e.g. by setting

The blowup can also be described by directly using coordinates on the normal space to the point.

Over the real or complex numbers, the blowup has a topological description as the connected sum

, preserves each line through the origin, and exchanges the inside of the sphere with the outside.

Let Pn - 1 be (n - 1)-dimensional complex projective space with homogeneous coordinates

be the subset of Cn × Pn - 1 that satisfies simultaneously the equations

) is called the blow-up (variously spelled blow up or blowup) of Cn.

The exceptional divisor E is defined as the inverse image of the blow-up locus Z under π.

If instead we consider the holomorphic projection we obtain the tautological line bundle of

This means that its normal bundle possesses no holomorphic sections;

is an invertible sheaf, characterized by this universal property: for any morphism f: Y → X such that

is the subscheme defined by the inverse image of the ideal sheaf

It follows from the definition of the blow up in terms of Proj that this subscheme E is defined by the ideal sheaf

In particular, in such cases the morphism π does not determine the exceptional divisor.

Fix a linear subspace L of codimension d. There are several explicit ways to describe the blowup of

Because L 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.

The following projective morphism of schemes gives a model of blowing up

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.

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 E. One must alter the symplectic form in a neighborhood of E, or perform the blow-up by cutting out a neighborhood of Z and collapsing the boundary in a well-defined way.