ring is commutative, reduced and finitely generated as a k-algebra; i.e., is the coordinate ring of an affine (reducible) variety X. For example, if A is the universal enveloping algebra of a finite-dimensional Lie algebra , then is a polynomial ring (the PBW theorem); more precisely, the coordinate ring of the dual vector space . Let M be a filtered module over A (i.e., ). If is fintiely generated as a -algebra, then the support of in X; i.e., the locus where does not vanish is called the characteristic variety of M. The notion plays an important role in the theory of D-modules. Projective variety A projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of homogeneous polynomials that generate a prime ideal. Example 1 A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The projective line P1 is an example of a projective curve; it can be viewed as the curve in the projective plane } defined by . For another example, first consider the affine cubic curve in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation: which defines a curve in P2 called an elliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P1, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves). Example 2: Grassmannian Let V be a finite-dimensional vector space. The Grassmannian variety Gn(V) is the set of all n-dimensional subspaces of V. It is a projective variety: it is embedded into a projective space via the Plücker embedding: where bi are any set of linearly independent vectors in V, is the n-th exterior power of V, and the bracket [w] means the line spanned by the nonzero vector w. The Grassmannian variety comes with a natural vector bundle (or locally free sheaf in other terminology) called the tautological bundle, which is important in the study of characteristic classes such as Chern classes. Jacobian variety and abelian variety Let C be a smooth complete curve and the Picard group of it; i.e., the group of isomorphism classes of line bundles on