More birational involutions

For $S$ a very general polarized K3 surface of degree $8n-6$, we describe in geometrical terms a birational involution of the Hilbert scheme $S^{[n]}$ of $n$ points on the surface, whose existence was established from lattice theoretical considerations. In a previous work we studied this involution for $n=3$, with the help of the exceptional Lie group $G_2$, since the Mukai model of $S$ is embedded in its projectivized Lie algebra. Here we use different, more general arguments to show that some important features of the birational involution persist for $n\ge 4$. In particular, we describe the indeterminacy locus of the involution in terms of a Mori contraction, and deduce that it is birational to a $\mathbb{P}^2$-fibration over a moduli space of sheaves on $S$, that also admits a degree two nef and big line bundle and an induced birational involution.