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  2. Rational mapping - Wikipedia

    en.wikipedia.org/wiki/Rational_mapping

    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).

  3. Morphism of schemes - Wikipedia

    en.wikipedia.org/wiki/Morphism_of_schemes

    For example, Spec k[x] and Spec k(x) and have the same function field (namely, k(x)) but there is no rational map from the former to the latter. However, it is true that any inclusion of function fields of algebraic varieties induces a dominant rational map (see morphism of algebraic varieties#Properties .)

  4. Birational geometry - Wikipedia

    en.wikipedia.org/wiki/Birational_geometry

    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.

  5. Morphism of algebraic varieties - Wikipedia

    en.wikipedia.org/wiki/Morphism_of_algebraic...

    If X is a smooth complete curve (for example, P 1) and if f is a rational map from X to a projective space P m, then f is a regular map X → P m. [5] In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P 1 and, conversely, such a morphism as a rational function on X.

  6. Algebraic geometry - Wikipedia

    en.wikipedia.org/wiki/Algebraic_geometry

    The domain of a rational function f is not V but the complement of the subvariety (a hypersurface) where the denominator of f vanishes. As with regular maps, one may define a rational map from a variety V to a variety V'. As with the regular maps, the rational maps from V to V' may be identified to the field homomorphisms from k(V') to k(V).

  7. Ample line bundle - Wikipedia

    en.wikipedia.org/wiki/Ample_line_bundle

    For example, if : is a dominant rational map between smooth projective varieties of the same dimension, then the pullback of a big line bundle on Y is big on X. (At first sight, the pullback is only a line bundle on the open subset of X where f is a morphism, but this extends uniquely to a line bundle on all of X .)

  8. List of complex and algebraic surfaces - Wikipedia

    en.wikipedia.org/wiki/List_of_complex_and...

    Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces; Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane

  9. Ruled variety - Wikipedia

    en.wikipedia.org/wiki/Ruled_variety

    For example, the conic x 2 + y 2 + z 2 = 0 in P 2 over the real numbers R is uniruled but not ruled. (The associated curve over the complex numbers C is isomorphic to P 1 and hence is ruled.) In the positive direction, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled.