 Blowing up

This article is about the mathematical concept of blowing up. For information about the physical/chemical process, see Explosion. For other uses of "Blow up", see Blow up (disambiguation).
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the 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 inflation of a balloon rather than an explosion.
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 all birational morphisms 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, σprocess, or Hopf map.
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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 G(1,2) parameterizes the set of all lines in the projective plane. The blowup of the projective plane at the point P, which we will denote X, is
X is a projective variety because it is a closed subvariety of a product of projective varieties. It comes with a natural morphism π to P^{2} that takes the pair to Q. This morphism is an isomorphism on the open subset of all points with Q ≠ P because the line is determined by those two points. When Q = P, however, the line can be any line through P. These lines correspond to the space of directions through P, which is isomorphic to P^{1}. This P^{1} is called the exceptional divisor, and by definition it is the projectivized normal space at P. Because P is a point, the normal space is the same as the tangent space, so the exceptional divisor is isomorphic to the projectivized tangent space at P.
To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P^{2} homogeneous coordinates [X_{0}:X_{1}:X_{2}] in which P is the point [P_{0}:P_{1}:P_{2}]. By projective duality, G(1,2) is isomorphic to P^{2}, so we may give it homogenous coordinates [L_{0}:L_{1}:L_{2}]. A line is the set of all [X_{0}:X_{1}:X_{2}] such that X_{0}L_{0} + X_{1}L_{1} + X_{2}L_{2} = 0. Therefore, the blowup can be described as
The blowup is an isomorphism away from P, and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation, we may assume that P = [0:0:1]. Write x and y for the coordinates on the affine plane X_{2}≠0. The condition P ∈ implies that L_{2} = 0, so we may replace the Grassmannian with a P^{1}. 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 also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane A^{2}. The normal space to the origin is the vector space m/m^{2}, where m = (x, y) 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
where x and y have degree 0 and z and w have degree 1.
Over the real or complex numbers, the blowup has a topological description as the connected sum . Assume that P is the origin in A^{2} ⊆ P^{2}, and write L for the line at infinity. A^{2} \ {0} has an inversion map t which sends (x, y) to (x/(x^{2} + y^{2}), y/(x^{2} + y^{2})). t is the circle inversion with respect to the unit sphere S: It fixes S, preserves each line through the origin, and exchanges the inside of the sphere with the outside. t extends to a continuous map P^{2} → A^{2} by sending the line at infinity to the origin. This extension, which we also denote t, can be used to construct the blowup. Let C denote the complement of the unit ball. The blowup X is the manifold obtained by attaching two copies of C along S. X comes with a map π to P^{2} which is the identity on the first copy of C and t on the second copy of C. This map is an isomorphism away from P, and the fiber over P is the line at infinity in the second copy of C. Each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin.
For CP^{2} this process ought to produce an oriented manifold. In order to make this happen, the two copies of C should be given opposite orientations. In symbols, X is , where is CP^{2} with the opposite of the standard orientation.
Blowing up points in complex space
Let Z be the origin in ndimensional complex space, C^{n}. That is, Z is the point where the n coordinate functions simultaneously vanish. Let P^{n  1} be (n  1)dimensional complex projective space with homogeneous coordinates . Let be the subset of C^{n} × P^{n  1} that satisfies simultaneously the equations x_{i}y_{j} = x_{j}y_{i} for . The projection
naturally induces a holomorphic map
This map π (or, often, the space ) is called the blowup (variously spelled blow up or blowup) of C^{n}.
The exceptional divisor E is defined as the inverse image of the blowup locus Z under π. It is easy to see that
is a copy of projective space. It is an effective divisor. Away from E, π is an isomorphism between and ; it is a birational map between and C^{n}.
Blowing up submanifolds in complex manifolds
More generally, one can blow up any codimensionk complex submanifold Z of C^{n}. Suppose that Z is the locus of the equations , and let be homogeneous coordinates on P^{k  1}. Then the blowup is the locus of the equations x_{i}y_{j} = x_{j}y_{i} for all i and j, in the space C^{n} × 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 blowup locus Z with the exceptional divisor E. In other words, the blowup map
is birational, and an isomorphism away from E. E is naturally seen as the projectivization of the normal bundle of Z. So is a locally trivial fibration with fiber 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 . (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 selfintersection 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 blowup . One is the proper (or strict) transform, which is the closure of ; its normal bundle in 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.
Blowing up schemes
To pursue blowup in its greatest generality, let X be a Noetherian scheme, and let be a coherent sheaf of ideals on X. The blowup of X with respect to is a scheme along with a morphism
such that is an invertible sheaf, characterized by this universal property: for any morphism such that is an invertible sheaf, f factors uniquely through π.
Notice that
has this property; this is how the blowup is constructed. Here Proj is the Proj construction on graded sheaves of commutative rings.
Exceptional divisors
The exceptional divisor of a blowup is the subscheme defined by the inverse image of the ideal sheaf , which is sometimes denoted . It follows from the definition of the blow up in terms of Proj that this subscheme E is defined by the ideal sheaf . This ideal sheaf is also the relative 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. This happens, for example, in the trivial situation where is the vanishing locus of a Cartier divisor. 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 P^{1} × P^{1}. X can be given as the vanishing locus of xw − yz in A^{4}. The ideals (x, y) and (x, z) 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.
Related constructions
In the blowup of C^{n} described above, there was nothing essential about the use of complex numbers; blowups can be performed over any field. For example, the real blowup of R^{2} at the origin results in the Möbius strip; correspondingly, the blowup of the twosphere results in the real projective plane.
Deformation to the normal cone is a blowup 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 blowup of X along V, and the other is the normal cone of V with its fibers completed to projective spaces.
Blowups can also be performed in the symplectic category, by endowing the symplectic manifold with a compatible almost complex structure and proceeding with a complex blowup. This makes sense on a purely topological level; however, endowing the blowup 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 blowup by cutting out a neighborhood of Z and collapsing the boundary in a welldefined way. This is best understood using the formalism of symplectic cutting, of which symplectic blowup 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
 Fulton, William (1998). Intersection Theory. SpringerVerlag. ISBN 0387985492.
 Griffiths, Phillip; Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0471327921.
 Hartshorne, Robin (1977). Algebraic Geometry. SpringerVerlag. ISBN 0387902449.
 McDuff, Dusa; Salamon, Dietmar (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 0198504519.
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