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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.
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.
For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., () is a ...
For a smooth complex projective variety, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory, Deligne cohomology. [7] This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian.
Affine space and projective space are smooth schemes over a field k. An example of a smooth hypersurface in projective space P n over k is the Fermat hypersurface x 0 d + ... + x n d = 0, for any positive integer d that is invertible in k. An example of a singular (non-smooth) scheme over a field k is the closed subscheme x 2 = 0 in the affine ...
Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C. [1] Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. [2]
An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by S .
A smooth curve together with a complete linear system of degree d > 2g is equivalent to a closed one dimensional subscheme of the projective space P d−g. Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space.