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 Cayley case $a=1$, we further construct an explicit rational polyhedral fundamental domain for the action of the automorphism group on the nef effective cone. For arbitrary $a$ and $n\ge\lfloor a^2/4\rfloor+3$, we determine the nef and Mori cones using explicit divisor classes whose nefness follows from the Mukai lattice description, together with curve classes obtained from pencils on smooth curves.