Given $X$ a k3 surface and $X^{[r]}$ the Hilbert scheme of points on it, I know that the Hilbert-Chow morphism $\rho:X^{[r]}\rightarrow X^{(r)}$ to the symmetric product is birational and, said $D\subset X^{(r)}$ the singular locus, $\rho^{-1}(D)$ is a divisor.
Beauville, in Varietes kahleriennes dont la premiere classe de chern est nulle (page 778), says that given $f:\mathcal{X}\rightarrow B$ a flat family of $X$, i can construct a flat family $g:\mathcal{Y}\rightarrow B$ in which the fibers are $\mathcal{Y}_b=\mathcal{X}_b^{[r]}$. I have some trouble seeing how it's done.
Of course taking the symmetric r-product and the desingularization of each fiber $\mathcal{X}_s$ the dimension of the fibers remains constant (so this checks out with the intuitive idea of flatness). But how can i prove that $g$ is proper and submersive?
