The Hilb-vs-Quot conjecture

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>y</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> yⁿ=xᵈ , where 𝑛 is either coprime to or divides 𝑑, we prove the Quot version of ORS through combinatorics. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n=3 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>3</m:mn> <m:mo>∤</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> 3 d , we deduce Hilb-vs-Quot by an asymptotic argument, and hence establish the original ORS Conjecture for these germs.