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In \cite{Vakil13} Vakil and Wood made several conjectures on the topology ofsymmetric powers of geometrically irreducible varieties based on theircomputations on motivic zeta functions. Two of those conjectures are aboutsubspaces of $\Sym^n(\P^1)$. In this note, we disprove one of them therebyobtaining a counterexample to the principle of Occcam's razor for Hodgestructures; and we prove that the other conjecture, with a minor correction,holds true.

On the cohomology of certain subspaces of $\mathit{Sym}^n(\P^1)$ and Occam's razor for Hodge structures