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In this case, the vanishing of. Therefore, projective varieties arise from homogeneous prime ideals I of k [ x 0 ,. Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X e. The explicit structure is as follows. 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. More generally,  projective space over a ring A is the union of the affine schemes. An equivalent but streamlined construction is given by the Proj construction , which is an analog of the spectrum of a ring , denoted "Spec", which defines an affine scheme.
Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. Moreover, there may be nilpotent functions on X. We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k , we let. A divisor D on a projective variety X corresponds to a line bundle L. One then set. Projective space over any scheme S can be defined as a fiber product of schemes.
Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. That "projective" implies "proper" is deeper: the main theorem of elimination theory. By definition, a variety is complete , if it is proper over k. The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing". There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete.
The converse is not true in general. In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below. Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete or projective. In fact, a projective subvariety of an affine variety must have dimension zero.
This is because only the constants are globally regular functions on a projective variety. By definition, any homogeneous ideal in a polynomial ring yields a projective scheme required to be prime ideal to give a variety. In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i. The product of two projective spaces is projective.
In fact, there is the explicit immersion called Segre embedding. As a consequence, the product of projective varieties over k is again projective.
As the prime ideal P defining a projective variety X is homogeneous, the homogeneous coordinate ring. It is a numerical invariant encoding some extrinsic geometry of X. The degree of P is the dimension r of X and its leading coefficient times r! If the homogeneous coordinate ring R is an integrally closed domain , then the projective variety X is said to be projectively normal.
Toroidal Groups Line Bundles Cohomology And Quasi Abelian Varieties 2001
Note, unlike normality , projective normality depends on R , the embedding of X into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of X. There are at least two equivalent ways to define the degree of X relative to its embedding. The first way is to define it as the cardinality of the finite set. This definition corresponds to an intuitive idea of a degree. Indeed, if X is a hypersurface, then the degree of X is the degree of the homogeneous polynomial defining X. The "general positions" can be made precise, for example, by intersection theory ; one requires that the intersection is proper and that the multiplicities of irreducible components are all one.
The other definition, which is mentioned in the previous section, is that the degree of X is the leading coefficient of the Hilbert polynomial of X times dim X!. Geometrically, this definition means that the degree of X is the multiplicity of the vertex of the affine cone over X. If m i denotes the multiplicity of an irreducible component Z i in the intersection i. If L is ample , then Proj of this ring is X. If the canonical ring is finitely generated, then Proj of the ring is called the canonical model of X. The canonical ring or model can then be used to define the Kodaira dimension of X.
Projective schemes of dimension one are called projective curves. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by normalization , which consists in taking locally the integral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of.
A smooth projective curve of genus one is called an elliptic curve. The Jacobian of a smooth curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curve E is E itself. For a curve X of genus g , Jac X has dimension g. Varieties, such as the Jacobian variety, which are complete and have a group structure are known as abelian varieties , in honor of Niels Abel. Moreover, they admit an ample line bundle and are thus projective.
On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces. Then the projection from E is the well-defined morphism. The geometric description of this map is as follows: . Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Thus, iterating the procedure, one sees there is a finite map.
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This result is the projective analog of Noether's normalization lemma. In fact, it yields a geometric proof of the normalization lemma. Fix a field k. An important case is when V consists of sections of a line bundle. Then there is a map: .
Related Toroidal Groups: Line Bundles, Cohomology and Quasi-Abelian Varieties
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