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A birational map from X to Y is a rational map f : X ⇢ Y such that there is a rational map Y ⇢ X inverse to f.A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by definition gives a birational map f : X ⇢ Y.
Formally, a rational map: between two varieties is an equivalence class of pairs (,) in which is a morphism of varieties from a non-empty open set to , and two such pairs (,) and (′ ′, ′) are considered equivalent if and ′ ′ coincide on the intersection ′ (this is, in particular, vacuously true if the intersection is empty, but since is assumed irreducible, this is impossible).
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 ...
birational morphism A birational morphism between schemes is a morphism that becomes an isomorphism after restricted to some open dense subset. One of the most common examples of a birational map is the map induced by a blowup. blow-up A blow-up is a
A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original version. The total transform of a normal point under a proper birational morphism is connected.
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety W with a proper birational map W→V. For varieties over fields of characteristic 0 , this was proved by Heisuke Hironaka in 1964; [ 1 ] while for varieties of dimension at least 4 over ...
Another name that exploded in popularity during the 1990s, Tyler is an English name with a literal meaning: "maker of tiles." In the 1990s, just over 262,000 Tylers were born in the United States ...
A generator of a scroll (ruled surface) that meets its consecutive generator. See (Semple & Roth 1949, p.204). torse Developable surface. transvectant An invariant depending on two forms. transversal A line meeting several other lines. For example, 4 generic lines in projective 3-space have 2 transversals meeting all of them. triad A set of 3 ...