canonical model of X. The canonical ring or model can then be used to define the Kodaira dimension of X. Projective curves Projective schemes of dimension one are called projective curves. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by normalization, which consists in taking locally the integral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of or equivalently smooth projective curves over is an important branch in algebraic number theory. A smooth projective curve of genus one is called an elliptic curve. As a consequence of the Riemann–Roch theorem, such a curve can be embedded as a closed subvariety in . In general, any (smooth) projective curve can be embedded in (for a proof, see Secant variety#Examples). Conversely, any smooth closed curve in of degree three has genus one by the genus formula and is thus an elliptic curve. A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism of degree two. Projective hypersurfaces Every irreducible closed subset of of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial. Abelian varieties Another important invariant of a projective variety X is the Picard group of X, the set of isomorphism classes of line bundles on X. It is isomorphic to and therefore an intrinsic notion (independent of embedding). For example, the Picard group of is isomorphic to via the degree map. The kernel of is not only an abstract abelian group, but there is a variety called the Jacobian variety of X, Jac(X), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curve E is E itself. For a curve X of genus g, Jac(X) has dimension g. Varieties, such as the Jacobian variety, which are complete and have a group structure are known as abelian varieties, in honor of Niels Abel. In marked contrast to affine algebraic groups such as , such groups are always commutative, whence the name. Moreover, they admit an ample line bundle and are thus projective. On the other hand, an abelian scheme may not be projective. Examples