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.
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 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) parametrizes the set of all lines through a point in the plane. The blowup of the projective plane P2 at the point P, which we will denote X, is
X=\{(Q,\ell)\midP,Q\in\ell\}\subseteqP2 x G(1,2).
\ell
(Q,\ell)
(Q,\ell)
\ell
\ell
To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P2 homogeneous coordinates [''X''<sub>0</sub>:''X''<sub>1</sub>:''X''<sub>2</sub>] in which P is the point [''P''<sub>0</sub>:''P''<sub>1</sub>:''P''<sub>2</sub>]. By projective duality, G(1,2) is isomorphic to P2, so we may give it homogeneous coordinates [''L''<sub>0</sub>:''L''<sub>1</sub>:''L''<sub>2</sub>]. A line
\ell0=[L0:L1:L2]
X=\{([X0:X1:X2],[L0:L1:L2])\midP0L0+P1L1+P2L2=0,X0L0+X1L1+X2L2=0\}\subseteqP2 x P2.
\ell
\{((x,y),[z:w])\midxz+yw=0\}\subseteqA2 x P1.
\left\{((x,y),[z:w])\mid\det\begin{bmatrix}x&y\\w&z\end{bmatrix}=0\right\}.
The blowup can be easily visualized if we remove the infinity point of the Grassmannian, e.g. by setting w = 1, and obtain the standard saddle surface y = xz in 3D space.
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,
X=\operatorname{Proj}
infty | |
oplus | |
r=0 |
r | |
\operatorname{Sym} | |
k[x,y] |
ak{m}/ak{m}2.
X=\operatorname{Proj}k[x,y][z,w]/(xz-yw),
P2\#P2
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
CP2\#\overline{CP2}
\overline{CP2}
Let Z be the origin in n-dimensional complex space, Cn. That is, Z is the point where the n coordinate functions
x1,\ldots,xn
y1,\ldots,yn
\tilde{Cn}
xiyj=xjyi
\pi:Cn x Pn\toCn
naturally induces a holomorphic map
\pi:\tilde{Cn}\toCn.
This map π (or, often, the space
\tilde{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
E=Z x Pn\subseteqCn x Pn
is a copy of projective space. It is an effective divisor. Away from E, π is an isomorphism between
\tilde{Cn}\setminusE
\tilde{Cn}
If instead we consider the holomorphic projection
q\colon\tilde{Cn}\toPn-1
Pn-1
\lbraceZ\rbrace x Pn-1
0\colonPn-1
\tol{O} | |
Pn-1 |
p
0p
p
More generally, one can blow up any codimension-k complex submanifold Z of Cn. Suppose that Z is the locus of the equations
x1= … =xk=0
y1,\ldots,yk
\tilde{C
xiyj=xjyi
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:\tildeX\toX
is a birational mapping which, away from E, induces an isomorphism, and, on E, a locally trivial fibration with fiber Pk - 1. Indeed, the restriction
\pi|E:E\toZ
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
\tildeX
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
\tildeX
\pi-1(V\setminusZ)
\tildeX
To pursue blow-up in its greatest generality, let X be a scheme, and let
l{I}
l{I}
\tilde{X}
\pi\colon\tilde{X} → X
such that
\pi-1l{I} ⋅ l{O}\tilde{X
f-1l{I} ⋅ l{O}Y
Notice that
\tilde{X}=Proj
infty | |
oplus | |
n=0 |
l{I}n
has this property; this is how the blow-up is constructed. Here Proj is the Proj construction on graded sheaves of commutative rings.
The exceptional divisor of a blowup
\pi:\operatorname{Bl}l{I}X\toX
l{I}
\pi-1l{I} ⋅ l{O}\operatorname{Bll{I}X}
infty | |
styleoplus | |
n=0 |
l{I}n+1
l{O}(1)
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
l{I}
Let
Pn
Pn
Pn
X0,...,Xn
L=\{Xn-d+1=...=Xn=0\}
Pn x Pn-d
Y0,...,Yn-d
P\inPn
Q\inPn-d
(P,Q)
This blowup can also be given a synthetic description as the incidence correspondencewhere
\operatorname{Gr}
(n-d+1)
Pn
M\in\operatorname{Gr}(n,n-d+1)
Pn-d
Pn
Pn-d
P\not\inL
(X0,...,Xn-d)
P\inL
(X0,...,Xn-d)
Let
f,g\inC[x,y,z]
d
d2
P2
d2
p=[x0:x1:x2]
f(p) ≠ 0
g(p) ≠ 0
P1
f(p)=g(p)=0
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
V x \{0\} in Y=X x C or X x P1
Then
\tildeY\toC
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.