Editing Projective variety (section)

{{Short description|Algebraic variety in a projective space}}
{{Dark mode invert|[[File:Addition on cubic (clean version).svg|thumb|An [[elliptic curve]] is a smooth projective curve of genus one.]]}}

In [[algebraic geometry]], a '''projective variety''' is an [[algebraic variety]] that is a closed [[Algebraic variety#Subvariety|subvariety]] of a [[projective space]]. That is, it is the zero-locus in <math>\mathbb{P}^n</math> of some finite family of [[homogeneous polynomial]]s that generate a [[prime ideal]], the defining ideal of the variety.

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]]

:<math>k[x_0, \ldots, x_n]/I</math>

is called the [[homogeneous coordinate ring]] of ''X''. Basic invariants of ''X'' such as the [[degree of an algebraic variety|degree]] and the [[dimension of an algebraic variety|dimension]] can be read off the [[Hilbert polynomial]] of this [[graded ring]].

Projective varieties arise in many ways. They are [[complete variety|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 bundle]]s or [[Divisor (algebraic geometry)|divisor]]s 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 of an algebraic variety|dimension]] 1. The theory of projective curves is particularly rich, including a classification by the [[Arithmetic genus|genus]] of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.<ref>{{harvnb|Kollár|Moduli|loc=Ch I.}}</ref> [[Hilbert scheme]]s parametrize closed subschemes of <math>\mathbb{P}^n</math> with prescribed Hilbert polynomial. Hilbert schemes, of which [[Grassmannian]]s 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 variety|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 number|complex]] coefficients. Broadly, the [[Algebraic geometry and analytic geometry|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 bundle]]s (more generally [[Coherent sheaf|coherent analytic sheaves]]) on ''X'' coincide with that of algebraic vector bundles. [[Algebraic geometry and analytic geometry#Chow's theorem|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]].