Nef Cones of the Hilbert Schemes of Points on Generalized Cayley K3 Surfaces

We study the nef cones of Hilbert schemes of points on generalized Cayley K3 surfaces $S_a$. For the Hilbert squares $S_a^{[2]}$ with $a=1,2$, we determine the nef and effective cones by combining Beauville involutions with the Bayer--Macrì wall description and explicit lattice calculations. In the