hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if X is a hypersurface, then the degree of X is the degree of the homogeneous polynomial defining X. The "general positions" can be made precise, for example, by intersection theory; one requires that the intersection is proper and that the multiplicities of irreducible components are all one. 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)!. Geometrically, this definition means that the degree of X is the multiplicity of the vertex of the affine cone over X. Let be closed subschemes of pure dimensions that intersect properly (they are in general position). If mi denotes the multiplicity of an irreducible component Zi in the intersection (i.e., intersection multiplicity), then the generalization of Bézout's theorem says: The intersection multiplicity mi can be defined as the coefficient of Zi in the intersection product in the Chow ring of . In particular, if 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. A complex projective variety can be viewed as a compact complex manifold; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambient complex projective space. A complex projective variety can be characterized as a minimizer of the volume (in a sense). The ring of sections Let X be a projective variety and L a line bundle on it. Then the graded ring is called the ring of sections of L. If L is ample, then Proj of this ring is X. Moreover, if X is normal and L is very ample, then is the integral closure of the homogeneous coordinate ring of X determined by L; i.e., so that pulls-back to L. For applications, it is useful to allow for divisors (or -divisors) not just line bundles; assuming X is normal, the resulting ring is then called a generalized ring of sections. If is a canonical divisor on X, then the generalized ring of sections is called the canonical ring of X. If the canonical ring is finitely generated, then Proj of the ring is called the