there are the notions of stable and semistable vector bundles on a smooth complete curve . The moduli of semistable vector bundles of a given rank and a given degree (degree of the determinant of the bundle) is then a projective variety denoted as , which contains the set of isomorphism classes of stable vector bundles of rank and degree as an open subset. Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of . In general, in contrast to the case of moduli of curves, a compactification of a moduli need not be unique and, in some cases, different non-equivalent compactifications are constructed using different methods and by different authors. An example over is the problem of compactifying , the quotient of a bounded symmetric domain by an action of an arithmetic discrete group . A basic example of is when , Siegel's upper half-space and commensurable with ; in that case, has an interpretation as the moduli of principally polarized complex abelian varieties of dimension (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify , a toroidal compactification of it. But there are other ways to compactify ; for example, there is the minimal compactification of due to Baily and Borel: it is the projective variety associated to the graded ring formed by modular forms (in the Siegel case, Siegel modular forms). The non-uniqueness of compactifications is due to the lack of moduli interpretations of those compactifications; i.e., they do not represent (in the category-theory sense) any natural moduli problem or, in the precise language, there is no natural moduli stack that would be an analog of moduli stack of stable curves. Non-affine and non-projective example An algebraic variety can be neither affine nor projective. To give an example, let and the projection. It is an algebraic variety since it is a product of varieties. It is not affine since P1 is a closed subvariety of X (as the zero locus of p), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety. It is not projective either, since there is a nonconstant regular function on X; namely, p. Another example of a non-affine non-projective variety is (cf. .) Non-examples Consider the affine line over . The complement of the circle