Hilbert scheme of points on non-reduced nodal curves

Abstract: We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $(\mathbb{C}^*)^{m_1} \times \mathbb{C}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincaré polynomials of the punctual Hilbert scheme of points on curves $x^uy^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1, v$ arbitrary, showing that the Poincaré polynomial is the row-colored link homology up to change of variables.

• The paper introduces new combinatorial stratifications for punctual Hilbert schemes on non-reduced nodal curves.
• It verifies the colored ORS conjecture for curves like {xy^v=0}, linking computed invariants to colored Khovanov-Rozansky homology.
• Explicit recursive formulas and affine paving techniques are developed, advancing computational methods in algebraic geometry and knot homologies.

Hilbert Schemes of Points on Non-Reduced Nodal Curves

Introduction and Background

The paper "Hilbert scheme of points on non-reduced nodal curves" (2604.03111) investigates a deep connection between algebraic geometry and low-dimensional topology, focusing on the moduli spaces of subschemes (Hilbert schemes) of singular plane curves with non-reduced structure, and their relationship to link homologies. Central to this study is the Oblomkov–Rasmussen–Shende (ORS) conjecture, which, in its colored version, relates the homology of Hilbert schemes of points localized at planar curve singularities to the colored Khovanov-Rozansky (KR) homology of algebraic links associated to those singularities.

Historically, the ORS conjecture has been verified in select cases involving reduced singular curves (notably $x^p=y^q$ for coprime $p,q$) and some non-reduced smooth curves. However, the case of non-reduced singular curves remained largely open, particularly for curves of the form $x^{u} y^{v}=0$ at the origin. This work provides explicit stratifications and computational techniques for the punctual Hilbert schemes of such curves, establishes affine pavings in several new cases, and verifies the colored ORS conjecture for strategically chosen non-reduced/singular examples.

Main Results

Hilbert Schemes and Colored Homology Correspondence

The primary achievement is the explicit classification of affine strata for punctual Hilbert schemes $Hilb^n(\{x^uy^v=0\},0)$. The paper provides:

• New stratifications via combinatorial data (the "weak diagonal partition"), enabling precise calculations of the homology and virtual Poincaré polynomials of these Hilbert schemes.
• Verification of the colored ORS conjecture for $u=1$, arbitrary $v$, specifically the case $C=\{xy^v=0\}$ yielding the Hopf link, with one component colored by a row of $v$ boxes and the other trivially colored. The calculated Poincaré polynomials coincide (up to an explicit change of variables) with the colored KR homologies derived from the recursion of Hogancamp-Mellit and Conners.

Structural and Combinatorial Advances

The approach circumvents the inadequacy of classical cell decompositions (such as Białynicki-Birula) for non-reduced singularities by introducing:

• Filtrations and non-reduced valuations tailored to the scheme structure of $x^u y^v=0$. These provide an effective means to catalog the possible generators of the ideals representing the Hilbert scheme points.
• Affine paving and torus-type stratification: For $u=v$, $p,q$0, $p,q$1, each stratum is described as $p,q$2, with dimensions determined by refined combinatorics on the weak diagonal partition. This leads to explicit formulas for virtual Poincaré polynomials and direct calculations for motivic invariants.

Explicit Computations

Key formulas include:

• For $p,q$3:

$p,q$4

Matching, up to variable change, the colored Poincaré polynomial of the Hopf link (see [HM] / [Con]) in the appropriate coloring.

• Generalizations for $p,q$5, arbitrary $p,q$6, involve more complex dependencies (e.g., stratification parameters based on the parity and partition data of the indexing).
• When $p,q$7, $p,q$8, or $p,q$9, virtual Poincaré polynomials are given recursively via adjacency matrices operating on vectors parameterized by the shape of the combinatorial partition. These are packaged in the explicit adjacency matrix recursion, yielding all motivic, cohomological, and numerical invariants recursively.

Stratification of the Punctual Hilbert Scheme of the Plane

For $x^{u} y^{v}=0$0, the scheme $x^{u} y^{v}=0$1 recovers $x^{u} y^{v}=0$2, providing a new, non-classical stratification of $x^{u} y^{v}=0$3. The corresponding virtual Poincaré polynomial recovers the classical generating series for integer partitions (Durfee square identity) as a limiting case.

Methodology

The technical core involves:

• A detailed recursive analysis of ideal generators for the punctual Hilbert scheme in the presence of non-reduced structure, via two types of filtrations (vertical and diagonal).
• Introduction of a non-reduced valuation that tracks leading terms compatible with the layer structure of the non-reduced curve.
• Classification of affine strata indexed by sequences of pairs $x^{u} y^{v}=0$4, subject to intricate inequalities reflecting the dependencies of generators across layers, encoded via the weak diagonal partition formalism.
• Comprehensive combinatorial enumeration using adjacency matrices, supporting both explicit rational expressions for motivic series and power series expansions via computer algebra (dedicated Macaulay2 code is provided).

Implications and Outlook

Theoretical Significance

• The verification of the colored ORS conjecture for non-reduced singular curves in new cases strengthens the conjectural bridge between Hilbert scheme geometry and link homology, advancing the categorification program for topological invariants.
• The robustness of the new stratifications (extending beyond Białynicki-Birula's regime) suggests a general method for singularities where the previous tools failed, with expected application to further classes of singular/non-reduced curves.
• The connection to motivic and virtual Poincaré polynomials, together with explicit combinatorial identities (including deformations of the Durfee formula), opens possibilities for further interplay with enumerative geometry and combinatorics.

Practical and Computational Aspects

• Computational techniques developed here, including the use of non-reduced valuations and recursive partition statistics, provide practical algorithms for calculating Betti, motivic, and cohomological invariants for arbitrary $x^{u} y^{v}=0$5, $x^{u} y^{v}=0$6 in small to intermediate cases.
• The explicit formulae serve as predictions for yet-unknown colored link homologies, suggesting that further advances in link homology computations may validate the conjectural equalities beyond $x^{u} y^{v}=0$7.

Future Directions

• Extending these methods to arbitrary $x^{u} y^{v}=0$8 remains a combinatorially challenging but theoretically promising task, with current results indicating that affine pavings (or, at least, virtual Poincaré polynomial computations) may require more sophisticated recursive stratifications or entirely new invariants.
• There is scope for developing general stratification techniques for Hilbert schemes on more complex (possibly non-planar, higher genus, or higher-dimensional) singular varieties.
• The interrelation with knot homologies for general links beyond Hopf links poses a compelling direction, particularly with respect to the colored and HOMFLY-PT invariants in link theory.

Conclusion

This work achieves a significant advance in the study of Hilbert schemes over non-reduced nodal curves by providing a detailed combinatorial and geometric stratification framework, verifying the colored ORS conjecture in new singular/non-reduced settings, and establishing explicit computational tools and combinatorial identities. The methods developed extend the range of curves and links where the interplay between algebraic geometry and link homology can be concretely realized. These results have substantial implications both for the theory of Hilbert schemes—illuminating new facets of their geometry over singular and non-reduced spaces—and for categorified knot invariants, providing explicit predictions and a blueprint for future verifications and generalizations.