to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree , so , we can construct a family of hypersurfaces defined as the vanishing locus of . Schematically, this can be constructed as using the projection we can see the fiber over a point is the projective hypersurface defined by . This can be used to show that the cycle class of every hypersurface of degree is rationally equivalent to , since can be used to establish a rational equivalence. Notice that the locus of is and it has multiplicity , which is the coefficient of its cycle class. Rational Equivalence of Cycles on a Curve If we take two distinct line bundles of a smooth projective curve , then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in . This is because for smooth varieties, so the divisor classes of and define inequivalent classes. The Chow ring When the scheme is smooth over a field , the Chow groups form a ring, not just a graded abelian group. Namely, when is smooth over , define to be the Chow group of codimension- cycles on . (When is a variety of dimension , this just means that .) Then the groups form a commutative graded ring with the product: The product arises from intersecting algebraic cycles. For example, if and are smooth subvarieties of of codimension and respectively, and if and intersect transversely, then the product in is the sum of the irreducible components of the intersection , which all have codimension . More generally, in various cases, intersection theory constructs an explicit cycle that represents the product in the Chow ring. For example, if and are subvarieties of complementary dimension (meaning that their dimensions sum to the dimension of ) whose intersection has dimension zero, then is equal to the sum of the points of the intersection with coefficients called intersection numbers. For any subvarieties and of a smooth scheme over , with no assumption on the dimension of the intersection, William Fulton and Robert MacPherson's intersection theory constructs a canonical element of the Chow groups of whose image in the Chow groups of is the product . Examples Projective space The Chow ring of projective space over any field is the ring where