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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.
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.
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 ...
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 ...
Another example of a singular variety is the projective cone of a smooth variety: ... The non-singularity of this scheme can also be checked using the Jacobian ...
These are all algebraic, and in some sense most surfaces are in this class. Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c 2 1 and c 2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However it ...
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.