is the class of a hyperplane (the zero locus of a single linear function). Furthermore, any subvariety of degree and codimension in projective space is rationally equivalent to . It follows that for any two subvarieties and of complementary dimension in and degrees , , respectively, their product in the Chow ring is simply where is the class of a -rational point in . For example, if and intersect transversely, it follows that is a zero-cycle of degree . If the base field is algebraically closed, this means that there are exactly points of intersection; this is a version of Bézout's theorem, a classic result of enumerative geometry. Projective bundle formula Given a vector bundle of rank over a smooth proper scheme over a field, the Chow ring of the associated projective bundle can be computed using the Chow ring of and the Chern classes of . If we let and the Chern classes of , then there is an isomorphism of rings Hirzebruch surfaces For example, the Chow ring of a Hirzebruch surface can be readily computed using the projective bundle formula. Recall that it is constructed as over . Then, the only non-trivial Chern class of this vector bundle is . This implies that the Chow ring is isomorphic to Remarks For other algebraic varieties, Chow groups can have richer behavior. For example, let be an elliptic curve over a field . Then the Chow group of zero-cycles on fits into an exact sequence Thus the Chow group of an elliptic curve is closely related to the group of -rational points of . When is a number field, is called the Mordell–Weil group of , and some of the deepest problems in number theory are attempts to understand this group. When is the complex numbers, the example of an elliptic curve shows that Chow groups can be uncountable abelian groups. Functoriality For a proper morphism of schemes over , there is a pushforward homomorphism for each integer . For example, for a proper scheme over , this gives a homomorphism , which takes a closed point in to its degree over . (A closed point in has the form for a finite extension field of , and its degree means the degree of the field over .) For a flat morphism of schemes over with fibers of dimension (possibly empty), there is a homomorphism . A key computational