Quot scheme of points on torus knot singularities

Abstract: For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[![T<sup>a,T<sup>b]!]$-submodules of $\Bbbk[![T]!]<sup>n$ is paved by affine cells, by proving that each Białynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}<em>m$-action is an affine bundle over the fixed point locus and that the fixed point locus is an iterated Grassmannian bundle. As an application, we determine the motive of this moduli space in the Grothendieck ring of varieties in terms of an explicit two-variable series $N</em>{a,b;n}(q,t)$, and use it to explicit compute the groupoid volume of the category of finite modules over $\mathbb{F}<em>q[![T<sup>a,T<sup>b]!]$. The series $N</em>{a,b;n}$ carries the conjectures we then formulate. At $n=\infty$ we conjecture a bi-infinite family of Rogers--Ramanujan type identities by specializing the $t$-variable; we identify their product side with the normalized character of a module over the $\mathcal{W}$-algebra minimal model $\mathcal{W}<em>a(a,a+b)$, and observe a connetion to colored Jones tails. At $n&lt;\infty$ we conjecture that $N</em>{a,b;n}$ is computed by the bottom $α$-row of the trigraded $S<sup>n$-colored HOMFLY homology of the torus knot $T(a,b)$, and that this same bottom row also computes the Quot schemes of finite codimensional $\Bbbk[![T<sup>a,T<sup>b]!]$-submoudles of $\Bbbk[![T<sup>a,T<sup>b]!]<sup>n$ and the punctual Hilbert schemes of the non-reduced curve $(Y<sup>a-X<sup>b)<sup>n=0$; the three quantities are special values at three points of the trigrading, and when $n=1$ they recover both the conjectures of Oblomkov--Rasmussen--Shende and of Kivinen--Trinh. Finally we conjecture that the one direction of the trigrading these three points do not see is a perverse filtration on the moduli spaces themselves, and we verify its prediction for a smooth germ at $n=2$ by computing the decomposition theorem for the $\mathrm{GL}_2$ spectral-curve family.

• The paper proves that high-rank punctual Quot schemes for torus knot singularities admit affine-cell decompositions, enabling explicit motivic generating functions for all ranks.
• Its proof combines tame gap-poset flag varieties, grid acyclicity, Gröbner strata, and Białynicki–Birula methods to control singular extension fibers and derive the polynomial N_{a,b;n}(q,t).
• The results establish finite groupoid volumes for modules over torus knot singularities and support conjectural links with colored HOMFLY homology, W-algebra characters, and Rogers–Ramanujan identities.

Overview and main results

This paper, by Huang, Jiang, and Oblomkov (2608.16086), studies the punctual Quot scheme $Quot_m^R(M)$ for the $(a,b)$-torus knot singularity $R=[\![T^a,T^b]\!]$ with $\gcd(a,b)=1$, in the high-rank regime $M=E^n$ where $E=[\![T]\!]$ or $R$. The rank-one case is classical: Poincaré polynomials of $Hilb_m(R)$ and $Quot^R_m([\![T]\!])$ are packaged by rational $q,t$-Catalan numbers, essentially the HOMFLY-PT polynomial of $(a,b)$0. The high-rank case had resisted explicit computation: torus actions yield little beyond Euler characteristics because of singularities, and $(a,b)$1-adic integration requires classifying torsion-free modules over $(a,b)$2, which is wild for most $(a,b)$3. Prior to this work only the $(a,b)$4 case was solved, exploiting finite CM type.

The central structural theorem asserts that for all $(a,b)$5, $(a,b)$6 admits a cell decomposition into affine spaces. Consequently its motivic generating function in the Grothendieck ring of varieties is

$(a,b)$7

where $(a,b)$8 and $(a,b)$9 is an explicit polynomial defined as a sum over vectors $R=[\![T^a,T^b]\!]$0 indexed by the gap set $R=[\![T^a,T^b]\!]$1, weighted by a generalized $R=[\![T^a,T^b]\!]$2-multinomial coefficient, a quadratic form $R=[\![T^a,T^b]\!]$3, and $R=[\![T^a,T^b]\!]$4.

Geometry of the extension fiber

The proof proceeds through the open subset $R=[\![T^a,T^b]\!]$5 of lattices whose normalization spans $R=[\![T^a,T^b]\!]$6. Under the natural $R=[\![T^a,T^b]\!]$7-action scaling $R=[\![T^a,T^b]\!]$8, the fixed point locus decomposes into components indexed by dimension vectors $R=[\![T^a,T^b]\!]$9, each a poset flag variety $\gcd(a,b)=1$0. Two general results drive the argument:

Poset flag varieties on tame posets. A poset is tame if it admits a "good traverse." For tame $\gcd(a,b)=1$1, the variety $\gcd(a,b)=1$2 is smooth projective, an iterated Grassmannian bundle over a point, affinely paved, with motive equal to a generalized $\gcd(a,b)=1$3-multinomial coefficient. The gap poset $\gcd(a,b)=1$4 embeds as a convex subposet (a Young diagram shape) of a rectangular grid, hence is tame. Poset flag varieties can be singular for non-tame posets — the paper exhibits a "butterfly" example with motive $\gcd(a,b)=1$5, violating Poincaré duality — so tameness is genuinely used.

Grid acyclicity. For a rectangular grid poset $\gcd(a,b)=1$6, injective $\gcd(a,b)=1$7-flag $\gcd(a,b)=1$8, and surjective $\gcd(a,b)=1$9-flag $M=E^n$0, the cochain complex

$M=E^n$1

is exact; equivalently $M=E^n$2. This is proved via Bardzell's resolution for quivers with relations, using surjectivity/injectivity of flag maps to solve lifting problems cell by cell. The authors note this is not a formal consequence of quiver theory since relations enter $M=E^n$3, and they exhibit configurations where exactness fails if $M=E^n$4's support degenerates.

The Białynicki-Birula fibers are Gröbner strata $M=E^n$5. Writing the condition that $M=E^n$6 be an $M=E^n$7-module as equations $M=E^n$8 ($M=E^n$9) in Gröbner coordinates, and solving degree by degree, solvability at each step reduces to a nonhomogeneous linear equation whose compatibility follows from commutativity of multiplication by $E=[\![T]\!]$0 and $E=[\![T]\!]$1 plus grid acyclicity applied after embedding the gap set into a grid. Each Białynicki-Birula stratum is thus an affine bundle of explicitly computable rank $E=[\![T]\!]$2 over $E=[\![T]\!]$3, with Zariski-local trivialization constructed elementarily (no Quillen–Suslin needed). A key combinatorial identity decouples the exponents:

$E=[\![T]\!]$4

proved by translating everything into Laurent-series convolution operators. This identity converts the motivic series into the clean form involving $E=[\![T]\!]$5, and relies crucially on the positive definiteness of $E=[\![T]\!]$6 on the cone $E=[\![T]\!]$7 (proved by the first author elsewhere), which guarantees finiteness phenomena downstream.

From extension fiber to Quot scheme and arithmetic consequences

Passing from $E=[\![T]\!]$8 to $E=[\![T]\!]$9 uses the Iwahori Schubert decomposition of the classical affine Grassmannian together with the constructible extension map $R$0, giving $R$1. This yields both the paving theorem and the motivic formula above.

For the free-module Quot scheme $R$2, the situation is harder: the paper does not know whether point counts are polynomial in $R$3 except when $R$4. However, combining the functional equation of Huang–Jiang with the evaluation $R$5 gives the sharp specialization

$R$6

Taking $R$7 via a monotone-convergence/Mertens pipeline, the paper proves that the groupoid volume of the category of finite modules over $R$8 equals a weighted count of commuting nilpotent matrix pairs $R$9 with $Hilb_m(R)$0:

$Hilb_m(R)$1

Finiteness rests on the positive definiteness theorem for $Hilb_m(R)$2, which the authors describe as an analytical miracle in this context. Consequently the infinite-rank Coh zeta function $Hilb_m(R)$3 has radius of convergence at least $Hilb_m(R)$4, resolving in the affirmative (at radius one) a weaker version of a question of the first author about infinite radius of convergence.

Conjectural framework: master polynomials and knot homology

The bulk of the conjectural material organizes three geometric quantities — $Hilb_m(R)$5, $Hilb_m(R)$6, and punctual Hilbert schemes of the thickened curve $Hilb_m(R)$7 — as slices of a single trivariate master polynomial $Hilb_m(R)$8 satisfying a functional equation, interpolation properties, cyclic sieving at roots of unity, and a Rogers–Ramanujan-type product expansion at infinite rank. In the unibranched case, the master polynomial is predicted to coincide with the bottom row of the reduced quadruply graded $Hilb_m(R)$9-colored HOMFLY homology of the algebraic knot, in GGS normalization:

$Quot^R_m([\![T]\!])$0

At $Quot^R_m([\![T]\!])$1 these recover the Oblomkov–Rasmussen–Shende and Kivinen–Trinh conjectures. The third variable $Quot^R_m([\![T]\!])$2 records a perverse filtration on the moduli spaces via virtual perverse weight polynomials; it supplies precisely the coordinate transverse to the two subtori seen by the unrefined slices.

On the product side, the charge function $Quot^R_m([\![T]\!])$3, describable as segment lengths cut by grid lines of an $Quot^R_m([\![T]\!])$4 rectangle along its diagonal, yields the conjecture $Quot^R_m([\![T]\!])$5, identified with the normalized character of the minimal-model $Quot^R_m([\![T]\!])$6-algebra module $Quot^R_m([\![T]\!])$7 labelled by the Euclidean rhythm $Quot^R_m([\![T]\!])$8. The $Quot^R_m([\![T]\!])$9 case recovers the Andrews–Gordon identities; the $q,t$0 case reduces to Warnaar's $q,t$1 identities. At the opposite specialization, $q,t$2 matches the Dirac-rhythm character, which appears as colored Jones tails of $q,t$3.

The knots–quivers correspondence serves as a computational model: rank-one data determine quiver vertices (indexed by order filters of the gap poset), linear gradings, and the diagonal of the symmetric matrix $q,t$4, but not off-diagonal entries. For $q,t$5 the explicit master-polynomial proposal reproduces the full KRSS matrix termwise. For the chain-like family $q,t$6 the complete matrix takes the simple form $q,t$7 on the diagonal and $q,t$8 off-diagonal, verified for $q,t$9, $(a,b)$00.

Evidence and verification

Finite-field enumeration confirms the Hilb-vs-Quot identity through colength 6 for $(a,b)$01 at $(a,b)$02, and the $(a,b)$03 coefficients agree with KRSS predictions through $(a,b)$04–$(a,b)$05 across several knots and ranks. Every stratum encountered was an affine space, so verified coefficients hold simultaneously as point counts and motivic identities. The strongest geometric evidence is Proposition on the ribbon at $(a,b)$06: for the split ribbon $(a,b)$07 in the $(a,b)$08 spectral-curve family over the Hitchin base, the relative Hilbert schemes are smooth through length 3, and the decomposition theorem computation — using the Chaudouard–Laumon support theorem and the Migliorini–Shende–Viviani formula, with the sign local system on the discriminant cone contributing anti-invariant node classes — verifies exactly the predicted perverse concentration, producing the generating function $(a,b)$09 up to $(a,b)$10. Notably, smoothness fails at length 4: the subscheme $(a,b)$11 is a singular point of the relative Hilbert scheme, so the smoothness hypothesis cannot be dropped globally.

Limitations and open questions

Several limitations are stated plainly. The paving result concerns $(a,b)$12 only; whether $(a,b)$13 has polynomial point counts for general $(a,b)$14 remains open, and no functional equation is known for $(a,b)$15 when $(a,b)$16. Whether every non-tame poset has some singular flag variety is unresolved. The closure structure of Schubert cells on the singular affine Grassmannian $(a,b)$17 — a Bruhat-type order — is posed as a question. The master polynomial conjecture assumes existence of interpolating polynomials for all plane curve germs, independence of the chosen deformation family for $(a,b)$18, purity and evenness of cohomology, and properness of the relevant families; none is proved in general. For multibranched singularities the predicted master polynomial can have negative coefficients (e.g., $(a,b)$19), precluding naive cohomological interpretation, and no link-theoretic model is available. Constructing the extremal quiver matrix $(a,b)$20 directly from Quot geometry — the off-diagonal entries being invisible to the two-variable slice — is stated as an open problem. Finally, the convergence of $(a,b)$21 beyond radius 1, and an arithmetic counterpart of the master framework interacting with Galois twists, remain open.

Conclusion

The paper establishes, unconditionally, that high-rank punctual Quot schemes of torus knot singularities are affinely paved, computes their motives through the explicit two-variable series $(a,b)$22, and derives the groupoid volume of finite modules over $(a,b)$23 together with convergence of the associated Coh zeta function at $(a,b)$24. The technical engine — grid acyclicity combined with a basis-free Gröbner deformation argument — replaces the finite-CM-type methods limited to $(a,b)$25 singularities. Around these theorems the authors build a coherent conjectural architecture connecting Quot schemes, thickened Hilbert schemes, perverse filtrations, colored HOMFLY homology, $(a,b)$26-algebra characters, and Rogers–Ramanujan type identities, supported by extensive finite-field and low-length computations but resting on several unproven structural hypotheses.