dimension k-1 and degree 1 is the projectivization of a k-dimensional subspace of n-dimensional affine space. This gives an isomorphism to a Grassmannian variety: The latter space has a distinguished system of homogeneous coordinates, given by the Plücker coordinates. Dimension 0: Points An effective algebraic cycle in of dimension 0 and degree d is an (unordered) d-tuple of points in , possibly with repetition. This gives an isomorphism to a symmetric power of : . Codimension 1: Divisors An effective algebraic cycle in of codimension 1 and degree d can be defined by the vanishing of a single degree d polynomial in n-many variables, and this polynomial is unique up to rescaling. Letting denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to a projective space: . Note that the latter space has a distinguished system of homogeneous coordinates, which send a polynomial to the coefficient of a fixed monomial. A non-trivial example The Chow variety parametrizes dimension 1, degree 2 cycles in . This Chow variety has two irreducible components. These two 8-dimensional components intersect in the moduli of coplanar pairs of lines, which is the singular locus in . This shows that, in contrast with the special cases above, Chow varieties need not be smooth or irreducible. The Chow embedding Let X be an irreducible subvariety in of dimension k-1 and degree d. By the definition of the degree, most -dimensional projective subspaces of intersect X in d-many points. By contrast, most -dimensional projective subspaces of do not intersect at X at all. This can be sharpened as follows. Lemma. The set parametrizing the subspaces of which intersect X non-trivially is an irreducible hypersurface of degree d. As a consequence, there exists a degree d form on which vanishes precisely on , and this form is unique up to scaling. This construction can be extended to an algebraic cycle by declaring that . To each degree d algebraic cycle, this associates a degree d form on , called the Chow form of X, which is well-defined up to scaling. Let denote the vector space of degree d forms on . The Chow-van-der-Waerden Theorem. The map which sends is a closed embedding of varieties. In particular, an effective algebraic cycle X is determined by its Chow form . If a basis for has been chosen, sending to the coefficients of in this basis