Chern-Schwartz-MacPherson classes of symmetric powers and applications

The Chern-Schwartz-MacPherson class $c_{SM}(X)$ is defined in the Chow group $A_*(X)$ of a scheme $X$, and generalizes $c(T_X)\cap [X]$ if $X$ is smooth. In this paper we independently compute $c_{SM}(S^nX)\in A_*(S^nX)$ modulo $n!$ torsion; we use this result to compute the pushforward to $S^nX$ of