Search results
Results from the WOW.Com Content Network
The fundamental group π 1 (X) is a birational invariant for smooth complex projective varieties. The "Weak factorization theorem", proved by Abramovich, Karu, Matsuki, and Włodarczyk , says that any birational map between two smooth complex projective varieties can be decomposed into finitely many blow-ups or blow-downs of smooth subvarieties ...
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.
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 ...
In the particular case that Y equals A 1 the regular maps f:X→A 1 are called regular functions, and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine ...
Print/export Download as PDF; Printable version; ... Pages in category "Birational geometry" The following 26 pages are in this category, out of 26 total. ...
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).
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
(Sometimes the induced birational morphism from to + is called a flip or flop.) In applications, is often a small contraction of an extremal ray, which implies several extra properties: The exceptional sets of both maps and + have codimension at least 2,