Examples of "birational"
is birational, the definition is extended by birational invariance.
In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence.
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.
Furthermore, one can select formula_18 that is birational with formula_19 and formula_20 that is birational with both formula_21 and formula_21 such that
The first example is given by the grounding work of Riemann himself: in his thesis, he shows that one can define a Riemann surface to each algebraic curve; every Riemann surface comes from an algebraic curve, well defined up to birational equivalence and two birational equivalent curves give the same surface. Therefore, the Riemann surface, or more simply its genus is a birational invariant.
is a birational equivalence from formula_115 to formula_46, with inverse:
Iskovskikh–Manin (1971) showed that the birational automorphism group of a smooth quartic 3-fold is equal to its automorphism group, which is finite. In this sense, quartic 3-folds are far from being rational, since the birational automorphism group of a rational variety is enormous. This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces.
A special case is a birational morphism "f": "X" → "Y", meaning a morphism which is birational. That is, "f" is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of "X" to points in "Y".
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry.
The maps formula_41 and formula_42 are birational mappings
The plurigenera are important birational invariants of an algebraic variety. In particular, the simplest way to prove that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus "P" with "d > 0"
Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup
Every algebraic variety is birational to a projective variety (Chow's lemma). So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting.
A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties. In other words, it depends only on the function field of the variety.
In these coordinates, the pentagram map is a birational mapping of formula_34
The fundamental group "π"("X") is a birational invariant for smooth complex projective varieties.
Weddle surfaces have 6 nodes and are birational to Kummer surfaces.
The group of birational automorphisms of the complex projective plane is the Cremona group.
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". In this case, "X" and "Y" are said to be birational, or birationally equivalent. In algebraic terms, two varieties over a field "k" are birational if and only if their function fields are isomorphic as extension fields of "k".
is birational map, the fibers of formula_24 are simply connected and the general fibers of formula_24