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. Examples and basic invariants 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.e., the case , is discussed further below. 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. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups. Homogeneous coordinate ring and Hilbert polynomial As the prime ideal P defining a projective variety X is homogeneous, the homogeneous coordinate ring is a graded ring, i.e., can be expressed as the direct sum of its graded components: There exists a polynomial P such that for all sufficiently large n; it is called the Hilbert polynomial of X. 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! is the degree of the variety X. The arithmetic genus of X is (−1)r (P(0) − 1) when X is smooth. For example, the homogeneous coordinate ring of is and its Hilbert polynomial is ; its arithmetic genus is zero. If the homogeneous coordinate ring R is an integrally closed domain, then the projective variety X is said to be projectively normal. 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. Degree Let be a projective variety. 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 where d is the dimension of X and Hi's are