Nef cone decompositions for nested Hilbert schemes of points on surfaces

Let $S$ be a smooth projective surface with $q(S)=0$. We give numerical criteria for the nef cones of $S^{[n,n+1]}$ and $S^{[1,n]}$ to decompose as sums of pullbacks of nef cones under their natural morphisms. These criteria recover the known decompositions for the projective plane, Hirzebruch surfaces, and Picard rank one K3 surfaces, and apply to all del Pezzo surfaces of degree at least two. For a del Pezzo surface of degree one, we show that the corresponding pullback decompositions fail for both $S^{[n,n+1]}$ and $S^{[1,n]}$.