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- Blowing_up abstract "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 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.".
- Blowing_up thumbnail Blowup.png?width=300.
- Blowing_up wikiPageID "1817790".
- Blowing_up wikiPageLength "17149".
- Blowing_up wikiPageOutDegree "48".
- Blowing_up wikiPageRevisionID "604036126".
- Blowing_up wikiPageWikiLink Almost_complex_manifold.
- Blowing_up wikiPageWikiLink Birational_geometry.
- Blowing_up wikiPageWikiLink Birational_morphism.
- Blowing_up wikiPageWikiLink Blowing_down.
- Blowing_up wikiPageWikiLink Cartier_divisor.
- Blowing_up wikiPageWikiLink Category:Birational_geometry.
- Blowing_up wikiPageWikiLink Category_theory.
- Blowing_up wikiPageWikiLink Circle_inversion.
- Blowing_up wikiPageWikiLink Coherent_sheaf.
- Blowing_up wikiPageWikiLink Cohomology.
- Blowing_up wikiPageWikiLink Complex_manifold.
- Blowing_up wikiPageWikiLink Complex_number.
- Blowing_up wikiPageWikiLink Connected_sum.
- Blowing_up wikiPageWikiLink Cremona_group.
- Blowing_up wikiPageWikiLink Divisor_(algebraic_geometry).
- Blowing_up wikiPageWikiLink Duality_(projective_geometry).
- Blowing_up wikiPageWikiLink Exceptional_divisor.
- Blowing_up wikiPageWikiLink Explosion.
- Blowing_up wikiPageWikiLink Fibration.
- Blowing_up wikiPageWikiLink Field_(mathematics).
- Blowing_up wikiPageWikiLink Graded_commutative_ring.
- Blowing_up wikiPageWikiLink Graded_ring.
- Blowing_up wikiPageWikiLink Grassmannian.
- Blowing_up wikiPageWikiLink Holomorphic_function.
- Blowing_up wikiPageWikiLink Homogeneous_coordinates.
- Blowing_up wikiPageWikiLink Homology_(mathematics).
- Blowing_up wikiPageWikiLink Incidence_correspondence.
- Blowing_up wikiPageWikiLink Infinitely_near_point.
- Blowing_up wikiPageWikiLink Inversive_geometry.
- Blowing_up wikiPageWikiLink Invertible_sheaf.
- Blowing_up wikiPageWikiLink Line_bundle.
- Blowing_up wikiPageWikiLink Mathematics.
- Blowing_up wikiPageWikiLink Möbius_strip.
- Blowing_up wikiPageWikiLink Normal_bundle.
- Blowing_up wikiPageWikiLink Normal_cone_(algebraic_geometry).
- Blowing_up wikiPageWikiLink Proj.
- Blowing_up wikiPageWikiLink Proj_construction.
- Blowing_up wikiPageWikiLink Projective_duality.
- Blowing_up wikiPageWikiLink Projective_plane.
- Blowing_up wikiPageWikiLink Projective_space.
- Blowing_up wikiPageWikiLink Real_projective_plane.
- Blowing_up wikiPageWikiLink Resolution_of_singularities.
- Blowing_up wikiPageWikiLink Scheme_(mathematics).
- Blowing_up wikiPageWikiLink Symplectic_cut.
- Blowing_up wikiPageWikiLink Symplectic_manifold.
- Blowing_up wikiPageWikiLink Symplectic_sum.
- Blowing_up wikiPageWikiLink Tangent_space.
- Blowing_up wikiPageWikiLink Universal_property.
- Blowing_up wikiPageWikiLink Weak_factorization_theorem.
- Blowing_up wikiPageWikiLink File:Blowup.png.
- Blowing_up wikiPageWikiLinkText "(real, oriented) blow-up".
- Blowing_up wikiPageWikiLinkText "Blowing up".
- Blowing_up wikiPageWikiLinkText "blow up".
- Blowing_up wikiPageWikiLinkText "blow-up".
- Blowing_up wikiPageWikiLinkText "blowing up".
- Blowing_up wikiPageWikiLinkText "blowing-up".
- Blowing_up wikiPageWikiLinkText "blown up".
- Blowing_up wikiPageWikiLinkText "blowup".
- Blowing_up wikiPageWikiLinkText "deformation to the normal cone".
- Blowing_up wikiPageWikiLinkText "monoidal transformation".
- Blowing_up hasPhotoCollection Blowing_up.
- Blowing_up wikiPageUsesTemplate Template:About.
- Blowing_up wikiPageUsesTemplate Template:Cite_book.
- Blowing_up subject Category:Birational_geometry.
- Blowing_up hypernym Transformation.
- Blowing_up type ProgrammingLanguage.
- Blowing_up type Variety.
- Blowing_up comment "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.".
- Blowing_up label "Blowing up".
- Blowing_up sameAs Éclatement_(mathématiques).
- Blowing_up sameAs 부풀리기.
- Blowing_up sameAs Expansão_(geometria_algébrica).
- Blowing_up sameAs m.05zbnz.
- Blowing_up sameAs Q1036595.
- Blowing_up sameAs Q1036595.
- Blowing_up sameAs 拉開.
- Blowing_up wasDerivedFrom Blowing_up?oldid=604036126.
- Blowing_up depiction Blowup.png.
- Blowing_up isPrimaryTopicOf Blowing_up.