Chow Ring - Geometric Interpretation

Geometric Interpretation

The geometric content of the Chow ring is the combination of rational equivalence and intersection product, which results in the seemingly formal numerical coefficients acquiring an interpretation in terms of the degree of a subvariety. For example, the Chow ring of projective space Pn can be shown to be:

where is the rational equivalence class of a hyperplane (the zero locus of a single linear functional). Furthermore, any subvariety Y of degree d and codimension k is rationally equivalent to, which means, for example, if we have two subvarieties Y and Z of complementary dimension (meaning their dimensions sum to n) and degrees d, e respectively, we get that their product is simply

where is the class of a point. This says, at least in the case when Y and Z intersect transversely (see below), that there are exactly de points of intersection; this is Bézout's theorem. Observations such as this, vastly generalized, give rise to the methods of enumerative geometry.