In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 at that point. The metaphor is inflation of a balloon

Balloon

A balloon is an inflatable flexible bag filled with a gas, such as helium, hydrogen, nitrous oxide, oxygen, or air. Modern balloons can be made from materials such as rubber, latex, polychloroprene, or a nylon fabric, while some early balloons were made of dried animal bladders, such as the pig...

 rather than an explosion

Explosion

An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...

.

Blowups are the most fundamental transformation in birational geometry

Birational geometry

In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...

, 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

Cremona group

In mathematics, in birational geometry, the Cremona group of order n over a field k is the group of birational automorphisms of the n-dimensional projective space over k...

, 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

Resolution of singularities

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...

 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

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 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

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

) 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

Grassmannian

In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

 G(1,2) parameterizes the set of all lines in the projective plane. The blowup of the projective plane

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

 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 P² 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 P¹. This P¹ is called the exceptional divisor

Exceptional divisor

In mathematics, specifically algebraic geometry, an exceptional divisor for a regular mapf: X \rightarrow Yof varieties is a kind of 'large' subvariety of X which is 'crushed' by f, in a certain definite sense...

, 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 P² homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

 [X₀:X₁:X₂] in which P is the point [P₀:P₁:P₂]. By projective duality, G(1,2) is isomorphic to P², so we may give it homogenous coordinates [L₀:L₁:L₂]. A line is the set of all [X₀:X₁:X₂] such that X₀L₀ + X₁L₁ + X₂L₂ = 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 X₂≠0. The condition P ∈ implies that L₂ = 0, so we may replace the Grassmannian with a P¹. 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 A². The normal space to the origin is the vector space m/m², 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

Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...

 . Assume that P is the origin in A² ⊆ P², and write L for the line at infinity. A² \ {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 P² → A² 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 P² 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 CP² 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 CP² with the opposite of the standard orientation.

Blowing up points in complex space

Let Z be the origin in n-dimensional complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 space, Cⁿ. 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 Cⁿ × P^{n - 1} that satisfies simultaneously the equations for . The projection


naturally induces a holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 map


This map π (or, often, the space ) 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

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...

. Away from E, π is an isomorphism between and ; it is a birational map between and Cⁿ.

Blowing up submanifolds in complex manifolds

More generally, one can blow up any codimension- complex submanifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

 Z of Cⁿ. Suppose that Z is the locus of the equations , and let be homogeneous coordinates on P^{k - 1}. Then the blow-up is the locus of the equations for all i and j, in the space Cⁿ × 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


is birational, and an isomorphism away from E. E is naturally seen as the projectivization of the normal bundle

Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...

 of Z. So is a locally trivial fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

 with fiber P^{k - 1}.

Since E is a smooth divisor, its normal bundle is a line bundle

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...

. 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

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 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 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 . 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

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

.

Blowing up schemes

To pursue blow-up in its greatest generality, let be a Noetherian

Noetherian

In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...

 scheme

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

, and let

be a coherent sheaf

Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...

 of ideals on . The blow-up of
with respect to is a scheme
along with a morphism

such that is an invertible sheaf

Invertible sheaf

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...

,
characterized by this universal property

Universal property

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

: for any morphism
such that is an invertible sheaf

Invertible sheaf

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...

, factors
uniquely through .

Notice that


has this property; this is how the blow-up is constructed. Here Proj is the Proj construction

Proj construction

In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties...

 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 .

is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of . That is, 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 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.

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

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

. For example, the real blow-up of R² at the origin results in the Möbius strip

Möbius strip

The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...

; correspondingly, the blow-up of the two-sphere results in the real projective plane

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

.

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

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

 with a compatible almost complex structure

Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...

 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 cut

Symplectic cut

In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut...

ting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of symplectic sum

Symplectic sum

In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum.The symplectic sum is...

mation, is the symplectic analogue of deformation to the normal cone along a smooth divisor.