Let $S$ be a scheme and $X$ be a scheme over $S$ corresponding to a smooth projective curve.
Define $X_d := X^d / \mathfrak{S}_d$, where $X^d$ denotes the usual $d$-fold cartesian product of $X$ and $\mathfrak{S}_d$ is the symmetric group on a set of $d$ elements.
Is it true that $X_d$ is a scheme over $S$, as well? If so, how can one show it?