In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. The inverse operation is called blowing down.

Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups.

Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by blowing up singularities until they become smooth. A consequence of this is that blowups can be used to resolve the singularities of birational maps.

Classically, blowups were defined extrinsically, by first defining the blowup on spaces such as projective space using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. This is reflected in some of the terminology, such as the classical term monoidal transformation. Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor.

A blowup can also be called monoidal transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map.

The blowup of a point in a plane

The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example.

The blowup has a synthetic description as an incidence correspondence. Recall that the Grassmannian parametrizes the set of all lines through a point in the plane. The blowup of the projective plane, is

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

homogeneous coordinates. By projective duality. Therefore, the blowup can be described as

, and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation. 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.

, and obtain the standard saddle surface in 3D space.

. The normal space to the origin is the vector space is the maximal ideal of the origin. Algebraically, the projectivization of this vector space is Proj of its symmetric algebra, that is,

In this example, this has a concrete description as

have degree have degree1.

Over the real or complex numbers, the blowup has a topological description as the connected sum for the line at infinity is the circle inversion corresponds to the possible normal directions through the origin.

with the opposite of the standard orientation.

Blowing up points in complex space

Let Z be the origin in n-dimensional complex space, Cn. That is, Z is the point where the n simultaneously vanish. Let Pn - 1 be (n be the subset of Cn Pn - 1 for i, j = 1, ..., n. The projection

naturally induces a holomorphic map

) 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. It is easy to see that

is a copy of projective space. It is an effective divisor. Away from E and Cn \ Z and Cn.

If instead we consider the holomorphic projection

we obtain the tautological line bundle.

Blowing up submanifolds in complex manifolds

complex submanifold.

. In other words, the blow-up map

, a locally trivial fibration is naturally seen as the projectivization of the normal bundle.

is a smooth divisor (which has co-dim 1), its normal bundle is a line bundle is the only smooth complex representative of its homology 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 .) This is why the divisor is called exceptional.

. One is the proper (or strict) transform. The other is the total transform in cohomology.

Blowing up schemes

To pursue blow-up in its greatest generality, let X be a scheme be a coherent sheaf of ideals on X. The blow-up of X along with a morphism

such that the pullback is an invertible sheaf, characterized by this universal property: for any morphism f: Y → X is an invertible sheaf, f factors uniquely through π.

Notice that

has this property; this is how the blow-up is constructed (see also Rees algebra). Here Proj is the Proj construction on graded sheaves of commutative rings.

Exceptional divisors

The exceptional divisor. It follows from the definition of the blow up in terms of Proj that this subscheme E. This ideal sheaf 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 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

be -dimensional projective space. Fix a linear subspace of dimension along be coordinates on the second factor. Because 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 no conditions on is in the blowup.

denotes the Grassmannian that contain . This is because each subspace is the linear join of and a point not in , and two points and determine the same away from , there is only one subspace containing , the linear join of and being the zero vector, and in this case, any is allowed, that is, any containing is possible.

Blowing up intersections of curves scheme-theoretically

points by Bézout's theorem). The following projective morphism of schemes.

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 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. 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.

See also

References