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Download as PDF; Printable version; In other projects ... In algebraic geometry, if : is a morphism of schemes, the fiber of a point in is the fiber product ...
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber ...
In algebraic geometry, the Stein factorization, introduced by Karl Stein for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.
For every object X, there exists a morphism id X : X → X called the identity morphism on X, such that for every morphism f : A → B we have id B ∘ f = f = f ∘ id A. Associativity h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever all the compositions are defined, i.e. when the target of f is the source of g, and the target of g is the source of h.
Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X × Y Spec(k(y)); this is a scheme over the field k(y). [3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface ...
A genus fibration: of is a proper flat morphism to a smooth curve such that and all fibers of have arithmetic genus. If X {\displaystyle X} is a smooth projective surface and the fibers of f {\displaystyle f} do not contain rational curves of self-intersection − 1 {\displaystyle -1} , then the fibration is called minimal .
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a P n -bundle if it is locally a projective n -space; i.e., X × S U ≃ P U n {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}} and transition automorphisms are linear.