Asymptotic behavior of the dimension of the Chow variety

First, we calculate the dimension of the Chow variety of degree d cycles on projective space. We show that the component of maximal dimension usually parametrizes degenerate cycles, confirming a conjecture of Eisenbud and Harris. Second, for a numerical class α on an arbitrary variety, we study how the dimension of the components of the Chow variety parametrizing cycles of class mα grows as we increase m. We show that when the maximal growth rate is achieved, α is represented by cycles that are “degenerate” in a precise sense.