Projective variety
Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
The converse is not true in general, but Chow's lemma describes the close relation of these two notions.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology.
The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.
, which can be defined in different, but equivalent ways: A projective variety is, by definition, a closed subvariety of
[2] In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions.
, the condition does not make sense for arbitrary polynomials, but only if f is homogeneous, i.e., the degrees of all the monomials (whose sum is f) are the same.
More generally,[4] projective space over a ring A is the union of the affine schemes in such a way the variables match up as expected.
That "projective" implies "proper" is deeper: the main theorem of elimination theory.
For example, for any projective variety X over k.[10] This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant).
The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely.
modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.
The first way is to define it as the cardinality of the finite set where d is the dimension of X and Hi's are hyperplanes in "general positions".
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)!.
be closed subschemes of pure dimensions that intersect properly (they are in general position).
is a hypersurface not containing X, then where Zi are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity (length of the local ring) mi.
Thus, iterating the procedure, one sees there is a finite map This result is the projective analog of Noether's normalization lemma.
The same procedure can be used to show the following slightly more precise result: given a projective variety X over a perfect field, there is a finite birational morphism from X to a hypersurface H in
parameterizes the lines in an affine n-space, the dual of it parametrizes the hyperplanes on the projective space, as follows.
on X satisfies the following important theorems due to Serre: These results are proven reducing to the case
In particular, if X is irreducible and has dimension r, the arithmetic genus of X is given by which is manifestly intrinsic; i.e., independent of the embedding.
Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ωX, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle.
For a (smooth projective) curve X, H2 and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional.
They will simply be called the genus of X. Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem.
A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods.
The converse is not in general true, but the Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective.
The fundamental Kodaira vanishing theorem states that for an ample line bundle
on a smooth projective variety X over a field of characteristic zero, for i > 0, or, equivalently by Serre duality
The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic.
Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.
