gives a system of homogeneous coordinates on the Chow variety , called the Chow coordinates of . However, as there is no consensus as to the ‘best’ basis for , this term can be ambiguous. From a foundational perspective, the above theorem is usually used as the definition of . That is, the Chow variety is usually defined as a subvariety of , and only then shown to be a fine moduli space for the moduli problem in question. Relation to the Hilbert scheme A more sophisticated solution to the problem of 'correctly' counting the degree of a degenerate subvariety is to work with subschemes of rather than subvarieties. Schemes can keep track of infinitesimal information that varieties and algebraic cycles cannot. For example, if two points in a variety approach each other in an algebraic family, the limiting subvariety is a single point, the limiting algebraic cycle is a point with multiplicity 2, and the limiting subscheme is a 'fat point' which contains the tangent direction along which the two points collided. The Hilbert scheme is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside . Each closed subscheme determines an effective algebraic cycle, and the induced map . is called the cycle map or the Hilbert-Chow morphism. This map is generically an isomorphism over the points in corresponding to irreducible subvarieties of degree d, but the fibers over non-simple algebraic cycles can be more interesting. Chow quotient A Chow quotient parametrizes closures of generic orbits. It is constructed as a closed subvariety of a Chow variety. Kapranov's theorem says that the moduli space of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian by the standard maximal torus. See also Picard variety GIT quotient References Mikhail Kapranov, Chow quotients of Grassmannian, I.M. Gelfand Seminar Collection, 29–110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. Algebraic geometry