In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring
is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann–Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.[1] Hilbert schemes parametrize closed subschemes of with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.
A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.
^Kollár & Moduli, Ch I.
and 27 Related for: Projective variety information
^{n}} . A projectivevariety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface...
called a projective algebraic set if V = Z(S) for some S.: 9 An irreducible projective algebraic set is called a projectivevariety.: 10 Projective varieties...
concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus...
in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore...
In algebraic geometry, a weighted projective space P(a0,...,an) is the projectivevariety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn]...
other words, a projectivevariety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the projective coordinates ring...
is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete...
embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projectivevariety. It is named after...
coming from any embedding into a projective space P n ( k ) {\displaystyle \mathbb {P} ^{n}(k)} as a quasi-projectivevariety. This is a group topology, and...
mathematics, the degree of an affine or projectivevariety of dimension n is the number of intersection points of the variety with n hyperplanes in general position...
Kähler manifolds. Hodge's primary motivation, the study of complex projectivevarieties, is encompassed by the latter case. Hodge theory has become an important...
Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. The original motivation...
bundle on X is equivalent to X being a projectivevariety, so a Fano variety is always projective. For a Fano variety X over the complex numbers, the Kodaira...
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that...
smooth projectivevarieties X. That is, this vector space is canonically identified with the corresponding space for any smooth projectivevariety which...
complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law...
complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...
coordinate ring of the variety. (Similarly, the normalization of a projectivevariety is a projectivevariety.) For an affine variety V ⊆ K n {\displaystyle...
distinctive feature of algebraic geometry is that some line bundles on a projectivevariety can be considered "positive", while others are "negative" (or a mixture...
vanish at a point of projective space. The projective Nullstellensatz gives a bijective correspondence between projectivevarieties and homogeneous ideals...
of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the...
numbers), every variety is birational to a smooth projectivevariety. Given that, it is enough to classify smooth projectivevarieties up to birational...
algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projectivevarieties – the...
which produces objects with the typical properties of projective spaces and projectivevarieties. The construction, while not functorial, is a fundamental...
Cox ring is a sort of universal homogeneous coordinate ring for a projectivevariety, and is (roughly speaking) a direct sum of the spaces of sections...
birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projectivevariety which is as simple as possible...