# Hilbert schemes of points on a Fano variety

Is the Hilbert scheme of points on a Fano variety a Fano variety itself?

The Hilbert scheme $S^{[n]}$ of $n$ points on a smooth surface $S$ is smooth for any $n$, but it is never Fano if $n>1$ since the canonical divisor is trivial on the fibers of the morphism to $S^{(n)}$, the $n$-th symmetric product. Indeed, the morphism from $S^{[n]}$ to $S^{(n)}$ is a crepant resolution of singularities. So neither the canonical nor the anticanonical line bundle can be ample on $S^{[n]}$.