Blowing up

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In mathematics, blowing up is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. At a point $Z$ that is being 'blown up' (the metaphor is inflation of a balloon, rather than explosion), $Z$ is replaced by the whole space of tangent directions at $Z$ (which, more formally, can be defined as the projective space constructed from the tangent space at $Z$ ). 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.

1 Blowing up points in complex space
2 Blowing up submanifolds in complex manifolds
3 Blowing up schemes
4 Related constructions
5 References

[edit] Blowing up points in complex space

Let $Z$ be the origin in $n$ -dimensional complex space, $\mathbb{C}^n$ . That is, $Z$ is the point where the $n$ coordinate functions $x_1, \ldots, x_n$ simultaneously vanish. Let $\mathbb{P}^{n - 1}$ be $(n - 1)$ -dimensional complex projective space with homogeneous coordinates $y_1, \ldots, y_n$ . Let $\tilde{\mathbb{C}^n}$ be the subset of $\mathbb{C}^n \times \mathbb{P}^{n - 1}$ that satisfies simultaneously the equations $x i y j = x j y i$ for $i, j = 1, \ldots, n$ . The projection

$\pi : \mathbb{C}^n \times \mathbb{P}^{n - 1} \to \mathbb{C}^n$

naturally induces a holomorphic map

$\pi : \tilde{\mathbb{C}^n} \to \mathbb{C}^n.$

This map $π$ (or, often, the space $\tilde{\mathbb{C}^n}$ ) is called the blow-up (variously spelled blow up or blowup) of $\mathbb{C}^n$ .

The exceptional divisor $E$ is defined as the inverse image of the blow-up locus $Z$ under $π$ . It is easy to see that

$E = Z \times \mathbb{P}^{n - 1} \subseteq \mathbb{C}^n \times \mathbb{P}^{n - 1}$

is a copy of projective space. It is an effective divisor. Away from $E$ , $π$ is an isomorphism between $\tilde{\mathbb{C}^n} \setminus E$ and $\mathbb{C}^n \setminus Z$ ; it is a birational map between $\tilde{\mathbb{C}^n}$ and $\mathbb{C}^n$ .

[edit] Blowing up submanifolds in complex manifolds

More generally, one can blow up any codimension- $k$ complex submanifold $Z$ of $\mathbb{C}^n$ . Suppose that $Z$ is the locus of the equations $x_1 = \cdots = x_k = 0$ , and let $y_1, \ldots, y_k$ be homogeneous coordinates on $\mathbb{P}^{k - 1}$ . Then the blow-up $\tilde{\mathbb{C}^n}$ is the locus of the equations $x i y j = x j y i$ for all $i$ and $j$ , in the space $\mathbb{C}^n \times \mathbb{P}^{k - 1}$ .

More generally still, one can blow up any submanifold of any complex manifold $X$ by applying this construction locally. The effect is, as before, to replace the blow-up locus $Z$ with the exceptional divisor $E$ . In other words, the blow-up map

$\pi : \tilde X \to X$

is birational, and an isomorphism away from $E$ . $E$ is naturally seen as the projectivization of the normal bundle of $Z$ . So $\pi|_E : E \to Z$ is a locally trivial fibration with fiber $\mathbb{P}^{k - 1}$ .

Since $E$ is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that $E$ intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; $E$ is the only smooth complex representative of its homology class in $\tilde X$ . (Suppose $E$ 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 $E$ .) This is why the divisor is called exceptional.

Let $V$ be some submanifold of $X$ other than $Z$ . If $V$ is disjoint from $Z$ , then it is essentially unaffected by blowing up along $Z$ . However, if it intersects $Z$ , then there are two distinct analogues of $V$ in the blow-up $\tilde X$ . One is the proper (or strict) transform, which is the closure of $\pi^{-1}(V \setminus Z)$ ; its normal bundle in $\tilde X$ is typically different from that of $V$ in $X$ . The other is the total transform, which incorporates some or all of $E$ ; it is essentially the pullback of $V$ in cohomology.

[edit] Blowing up schemes

To pursue blow-up in its greatest generality, let $X$ be a Noetherian scheme, and let $\mathcal{I}$ be a coherent sheaf of ideals on $X$ . The blow-up of $X$ with respect to $\mathcal{I}$ is a scheme $\tilde{X}$ along with a morphism

$\pi: \tilde{X} \rightarrow X$

such that $\pi^{-1} \mathcal{I} \cdot \mathcal{O}_Y$ is an invertible sheaf, characterized by this universal property: for any morphism $f: Y \rightarrow X$ such that $f^{-1} \mathcal{I} \cdot \mathcal{O}_Y$ is an invertible sheaf, $f$ factors uniquely through $π$ .

Notice that

$\tilde{X}=\mathbf{Proj} \oplus_{n=0}^{\infty} \mathcal{I}^n$

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

[edit] Related constructions

In the blow-up of $\mathbb{C}^n$ 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 $\mathbb{R}^2$ at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere $\mathbb{S}^2$ 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 $X$ and a closed subscheme $V$ , one blows up $V \times \{0\}$ in $Y = X \times \mathbb{C}$ (or $X \times \mathbb{P}^1$ ). Then

$\tilde Y \to X \times \mathbb{C}$

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.

[edit] 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.