C. Since C is smooth, can be identified as the divisor class group of C and thus there is the degree homomorphism . The Jacobian variety of C is the kernel of this degree map; i.e., the group of the divisor classes on C of degree zero. A Jacobian variety is an example of an abelian variety, a complete variety with a compatible abelian group structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (theta functions in the algebraic setting gives an embedding); thus, is a projective variety. The tangent space to at the identity element is naturally isomorphic to hence, the dimension of is the genus of . Fix a point on . For each integer , there is a natural morphism where is the product of n copies of C. For (i.e., C is an elliptic curve), the above morphism for turns out to be an isomorphism; in particular, an elliptic curve is an abelian variety. Moduli varieties Given an integer , the set of isomorphism classes of smooth complete curves of genus is called the moduli of curves of genus and is denoted as . There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use geometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure. Moduli such as the moduli of curves of fixed genus is typically not a projective variety; roughly the reason is that a degeneration (limit) of a smooth curve tends to be non-smooth or reducible. This leads to the notion of a stable curve of genus , a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves , the set of isomorphism classes of stable curves of genus , is then a projective variety which contains as an open subset. Since is obtained by adding boundary points to , is colloquially said to be a compactification of . Historically a paper of Mumford and Deligne introduced the notion of a stable curve to show is irreducible when . The moduli of curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here,