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We also construct a sheaf on , called the “structure sheaf” as in the affine case, which makes it into a scheme.As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following.
The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
An object in a category is projective if for any epimorphism: and morphism:, there is a morphism ¯: such that ¯ =, i.e. the following diagram commutes: That is, every morphism P → X {\displaystyle P\to X} factors through every epimorphism E ↠ X {\displaystyle E\twoheadrightarrow X} .
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e., there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points.
For m = 0 the generalized Jacobian J m is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group L m of dimension deg(m)−1. So we have an exact sequence 0 → L m → J m → J → 0
In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme.More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a coherent sheaf on X, then there is a scheme whose set of T-points () = (, ()) is the set of isomorphism classes of the quotients of that are flat over T.
Jouanolou's original statement was: If X is a scheme quasi-projective over an affine scheme, then there exists a vector bundle E over X and an affine E-torsor W.. By the definition of a torsor, W comes with a surjective map to X and is Zariski-locally on X an affine space bundle.