Hilbert schemes of points on a Fano variety

Question

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

Answer 1

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 $S^{[n]}$.

Answer 2

The Hilbert scheme of points on a Fano variety of dimension at least 3 is not even a variety if the number of points is sufficiently large. Also it has bad singularities where components of different dimensions meet, which makes it hard to define the anticanonical sheaf so it is not even clear what "Fano" might mean for such a scheme.