Blowing up

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 Failed to parse (Missing texvc executable; please see math/README to configure.): Z

that is being 'blown up' (the metaphor is inflation of a balloon, rather than explosion), Failed to parse (Missing texvc executable; please see math/README to configure.): Z
is replaced by the whole space of tangent directions at Failed to parse (Missing texvc executable; please see math/README to configure.): Z
(which, more formally, can be defined as the projective space constructed from the tangent space at Failed to parse (Missing texvc executable; please see math/README to configure.): Z

). 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.

Blowing up points in complex space

Let Failed to parse (Missing texvc executable; please see math/README to configure.): Z

be the origin in Failed to parse (Missing texvc executable; please see math/README to configure.): n

-dimensional complex space, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n . That is, Failed to parse (Missing texvc executable; please see math/README to configure.): Z

is the point where the Failed to parse (Missing texvc executable; please see math/README to configure.): n
coordinate functions Failed to parse (Missing texvc executable; please see math/README to configure.): x_1, \ldots, x_n
simultaneously vanish. Let Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^{n - 1}
be Failed to parse (Missing texvc executable; please see math/README to configure.): (n - 1)

-dimensional complex projective space with homogeneous coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): y_1, \ldots, y_n . Let Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{\mathbb{C}^n}

be the subset of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n \times \mathbb{P}^{n - 1}
that satisfies simultaneously the equations Failed to parse (Missing texvc executable; please see math/README to configure.): x_i y_j = x_j y_i 
for Failed to parse (Missing texvc executable; please see math/README to configure.): i, j = 1, \ldots, n

. The projection

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi : \mathbb{C}^n \times \mathbb{P}^{n - 1} \to \mathbb{C}^n

naturally induces a holomorphic map

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi : \tilde{\mathbb{C}^n} \to \mathbb{C}^n.

This map Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

(or, often, the space Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{\mathbb{C}^n}

) is called the blow-up (variously spelled blow up or blowup) of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n .

The exceptional divisor Failed to parse (Missing texvc executable; please see math/README to configure.): E

is defined as the inverse image of the blow-up locus Failed to parse (Missing texvc executable; please see math/README to configure.): Z
under Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

. It is easy to see that

Failed to parse (Missing texvc executable; please see math/README to configure.): E = Z \times \mathbb{P}^{n - 1} \subseteq \mathbb{C}^n \times \mathbb{P}^{n - 1}

is a copy of projective space. It is an effective divisor. Away from Failed to parse (Missing texvc executable; please see math/README to configure.): E , Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

is an isomorphism between Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{\mathbb{C}^n} \setminus E
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n \setminus Z

it is a birational map between Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{\mathbb{C}^n}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n

.

Blowing up submanifolds in complex manifolds

More generally, one can blow up any codimension-Failed to parse (Missing texvc executable; please see math/README to configure.): k

complex submanifold Failed to parse (Missing texvc executable; please see math/README to configure.): Z
of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n

. Suppose that Failed to parse (Missing texvc executable; please see math/README to configure.): Z

is the locus of the equations Failed to parse (Missing texvc executable; please see math/README to configure.): x_1 = \cdots = x_k = 0

, and let Failed to parse (Missing texvc executable; please see math/README to configure.): y_1, \ldots, y_k

be homogeneous coordinates on Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^{k - 1}

. Then the blow-up Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{\mathbb{C}^n}

is the locus of the equations Failed to parse (Missing texvc executable; please see math/README to configure.): x_i y_j = x_j y_i
for all Failed to parse (Missing texvc executable; please see math/README to configure.): i
and Failed to parse (Missing texvc executable; please see math/README to configure.): j

, in the space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n \times \mathbb{P}^{k - 1} .

More generally still, one can blow up any submanifold of any complex manifold Failed to parse (Missing texvc executable; please see math/README to configure.): X

by applying this construction locally. The effect is, as before, to replace the blow-up locus Failed to parse (Missing texvc executable; please see math/README to configure.): Z
with the exceptional divisor Failed to parse (Missing texvc executable; please see math/README to configure.): E

. In other words, the blow-up map

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi : \tilde X \to X

is birational, and an isomorphism away from Failed to parse (Missing texvc executable; please see math/README to configure.): E . Failed to parse (Missing texvc executable; please see math/README to configure.): E

is naturally seen as the projectivization of the normal bundle of Failed to parse (Missing texvc executable; please see math/README to configure.): Z

. So Failed to parse (Missing texvc executable; please see math/README to configure.): \pi|_E : E \to Z

is a locally trivial fibration with fiber Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^{k - 1}

.

Since Failed to parse (Missing texvc executable; please see math/README to configure.): E

is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that Failed to parse (Missing texvc executable; please see math/README to configure.): E
intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; Failed to parse (Missing texvc executable; please see math/README to configure.): E
is the only smooth complex representative of its homology class in Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde X

. (Suppose Failed to parse (Missing texvc executable; please see math/README to configure.): E

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 Failed to parse (Missing texvc executable; please see math/README to configure.): E

.) This is why the divisor is called exceptional.

Let Failed to parse (Missing texvc executable; please see math/README to configure.): V

be some submanifold of Failed to parse (Missing texvc executable; please see math/README to configure.): X
other than Failed to parse (Missing texvc executable; please see math/README to configure.): Z

. If Failed to parse (Missing texvc executable; please see math/README to configure.): V

is disjoint from Failed to parse (Missing texvc executable; please see math/README to configure.): Z

, then it is essentially unaffected by blowing up along Failed to parse (Missing texvc executable; please see math/README to configure.): Z . However, if it intersects Failed to parse (Missing texvc executable; please see math/README to configure.): Z , then there are two distinct analogues of Failed to parse (Missing texvc executable; please see math/README to configure.): V

in the blow-up Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde X

. One is the proper (or strict) transform, which is the closure of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi^{-1}(V \setminus Z)

its normal bundle in Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde X

is typically different from that of Failed to parse (Missing texvc executable; please see math/README to configure.): V
in Failed to parse (Missing texvc executable; please see math/README to configure.): X

. The other is the total transform, which incorporates some or all of Failed to parse (Missing texvc executable; please see math/README to configure.): E

it is essentially the pullback of Failed to parse (Missing texvc executable; please see math/README to configure.): V

in cohomology.

Blowing up schemes

To pursue blow-up in its greatest generality, let Failed to parse (Missing texvc executable; please see math/README to configure.): X

be a Noetherian scheme, and let

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{I}

be a coherent sheaf of ideals on Failed to parse (Missing texvc executable; please see math/README to configure.): X . The blow-up of Failed to parse (Missing texvc executable; please see math/README to configure.): X

with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{I}
is a scheme

Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{X}

along with a morphism

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi\colon \tilde{X} \rightarrow X

such that Failed to parse (Missing texvc executable; please see math/README to configure.): \pi^{-1} \mathcal{I} \cdot \mathcal{O}_{\tilde{X}}

is an invertible sheaf,

characterized by this universal property: for any morphism Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon Y \rightarrow X

such that Failed to parse (Missing texvc executable; please see math/README to configure.): f^{-1} \mathcal{I} \cdot \mathcal{O}_Y
is an invertible sheaf, Failed to parse (Missing texvc executable; please see math/README to configure.): f
factors

uniquely through Failed to parse (Missing texvc executable; please see math/README to configure.): \pi .

Notice that

Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{X}=\mathbf{Proj} \bigoplus_{n=0}^{\infty} \mathcal{I}^n

has this property; this is how the blow-up is constructed. Here Proj is the Proj construction on graded commutative rings.

Related constructions

In the blow-up of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^n

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^2
at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{S}^2
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 Failed to parse (Missing texvc executable; please see math/README to configure.): X

and a closed subscheme Failed to parse (Missing texvc executable; please see math/README to configure.): V

, one blows up Failed to parse (Missing texvc executable; please see math/README to configure.): V \times \{0\}

in Failed to parse (Missing texvc executable; please see math/README to configure.): Y = X \times \mathbb{C}
(or Failed to parse (Missing texvc executable; please see math/README to configure.): X \times \mathbb{P}^1

). Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde Y \to X \times \mathbb{C}

is a fibration. The general fiber is naturally isomorphic to Failed to parse (Missing texvc executable; please see math/README to configure.): X , while the central fiber is a union of two schemes: one is the blow-up of Failed to parse (Missing texvc executable; please see math/README to configure.): X

along Failed to parse (Missing texvc executable; please see math/README to configure.): V

, and the other is the normal cone of Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): E . One must alter the symplectic form in a neighborhood of Failed to parse (Missing texvc executable; please see math/README to configure.): E , or perform the blow-up by cutting out a neighborhood of Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

References

• Fulton, William (1998). Intersection Theory. Springer-Verlag. ISBN 0-387-98549-2. 
• Griffiths, Phillip and Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0-471-32792-1. 
• Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9. 
• McDuff, Dusa and Salamon, Dietmar (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 0-19-850451-9. 

See also

• Blowing down