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On the cohomology of certain subspaces of \(\mathrm{Sym}^n(\mathbb{P}^1)\) and Occam’s razor for Hodge structures. (English) Zbl 1507.14015

Summary: R. Vakil and M. Matchett-Wood [“Discriminants in the Grothendieck ring of varieties”, Preprint, arXiv:1208.3166] made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of \(\mathrm{Sym}^n(\mathbb{P}^1)\). In this note, we disprove one of them and prove a stronger form of the other, thereby obtaining (counter)examples to the principle of Occam’s razor for Hodge structures.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
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