Abstract Let 𝑅 be the complete local ring of a complex plane curve germ and 𝑆 its normalization. We propose a “Hilb-vs-Quot” conjecture relating the virtual weight polynomials of the Hilbert schemes of 𝑅 to those of the Quot schemes that parametrize 𝑅-submodules of 𝑆. By relating the Quot side to a type of compactified Picard scheme, we show that our conjecture generalizes a conjecture of Cherednik’s, and that it would relate the perverse filtration on the cohomology of the Picard side to a more elementary filtration. Next, we propose a Quot version of the Oblomkov–Rasmussen–Shende Conjecture, relating parabolic refinements of our Quot schemes to Khovanov–Rozansky link homology. It becomes equivalent to the original version under (refined) Hilb-vs-Quot, but can also be strengthened to incorporate polynomial actions and 𝑦-ification. For germs y n = x d y^n=x^d, where 𝑛 is either coprime to or divides 𝑑, we prove the Quot version of ORS through combinatorics. When n = 3 n=3 and 3 ∤ d 3 d, we deduce Hilb-vs-Quot by an asymptotic argument, and hence establish the original ORS Conjecture for these germs.
Kivinen et al. (Wed,) studied this question.
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