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In algebraic geometry , a projective variety over an algebraically closed field k is a subset of some projective n-space Pn 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 .. algebraic geometry. algebraic geometry. algebraically closed field. algebraically closed field. prime ideal. prime ideal. Equivalently , an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of Pn .. algebraic variety. algebraic variety. Zariski closed. Zariski topology. subvariety. Algebraic variety#Subvariety. 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 .. homogeneous polynomial. homogeneous polynomial. dimension. dimension of an algebraic variety. If X is a projective variety defined by a homogeneous prime ideal I , then the quotient ring. prime ideal. prime ideal. quotient ring. 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 .. homogeneous coordinate ring. homogeneous coordinate ring. degree. degree of an algebraic variety. dimension. dimension of an algebraic variety. Hilbert polynomial. Hilbert polynomial. graded ring. graded ring. Projective varieties arise in many ways .. They are complete , which roughly can be expressed by saying that there are no points `` missing '' .. complete. complete variety. 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 Poincare duality .. Serre duality. Serre duality. Poincare duality. Poincare duality. It also leads to the Riemann-Roch theorem for projective curves , i.e. , projective varieties of dimension 1 .. dimension. dimension of an algebraic variety. Riemann-Roch theorem. Riemann-Roch theorem. The theory of projective curves is particularly rich , including a classification by the genus of the curve .. genus. Arithmetic genus. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties .. Hilbert schemes parametrize closed subschemes of Pn with prescribed Hilbert polynomial .. Hilbert polynomial. Hilbert polynomial. Hilbert schemes , of which Grassmannians are special cases , are also projective schemes in their own right .. Geometric invariant theory offers another approach .. Geometric invariant theory. Geometric invariant theory. The classical approaches include the Teichmuller space and Chow varieties .. Teichmuller space. Teichmuller space. Chow varieties. Chow variety. 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 .. complex. complex number. Broadly , the GAGA principle says that the geometry of projective complex analytic spaces -LRB- or manifolds -RRB- is equivalent to the geometry of projective complex varieties .. complex. complex number. GAGA principle. Algebraic geometry and analytic geometry. For example , the theory of holomorphic vector bundles -LRB- more generally coherent analytic sheaves -RRB- on X coincide with that of algebraic vector bundles .. coherent analytic sheaves. Coherent sheaf. 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 .. complex. complex number. Hodge theory. Hodge theory