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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with 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 mathematics, a projective space is a fundamental construction from any vector space. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing 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. ...

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. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...

Blowing up points in complex space

Let Z be the origin in n-dimensional complex space, . That is, Z is the point where the n coordinate functions simultaneously vanish. Let be (n − 1)-dimensional complex projective space with homogeneous coordinates . Let be the subset of that satisfies simultaneously the equations x_iy_j = x_jy_i for . The projection Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ...

naturally induces a holomorphic map Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

This map π (or, often, the space ) is called the blow-up (variously spelled blow up or blowup) of .

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. Away from E, π is an isomorphism between and ; it is a birational map between and . In algebraic geometry, divisors are a generalization of subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). ...

Blowing up submanifolds in complex manifolds

More generally, one can blow up any codimension-k complex submanifold Z of . Suppose that Z is the locus of the equations , and let be homogeneous coordinates on . Then the blow-up is the locus of the equations x_iy_j = x_jy_i for all i and j, in the space . In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

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 of Z. So is a locally trivial fibration with fiber . In the mathematical field of differential geometry, a normal bundle is a particular kind of vector bundle. ... This article may be too technical for most readers to understand. ...

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 self-intersection of E.) This is why the divisor is called exceptional. In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

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. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

Blowing up schemes

To pursue blow-up in its greatest generality, let X be a Noetherian scheme, and let be a coherent sheaf of ideals on X. The blow-up of X with respect to is a scheme along with a morphism This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ...

characterized by this universal property: for any morphism such that is an invertible sheaf, f factors uniquely through π. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... 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. ...

Notice that

has this property; this is how the blow-up is constructed. Here Proj is the Proj construction on graded commutative rings. Proj is a certain construction in mathematics, more precisely in the field of algebraic geometry. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

Related constructions

In the blow-up of described above, there was nothing essential about the use of complex numbers; blow-ups can be performed over any field. For example, the blow-up of at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere results in the real projective plane. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The Möbius strip or Möbius band (named after the German mathematician and astronomer August Ferdinand Möbius) is a topological object with only one surface and only one edge. ... In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ...

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 in (or ). 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 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. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ...

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, Inc.. 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 0-198-50451-9.