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Invariants of nested Hilbert and Quot schemes on surfaces

Nadir Fasola, Michele Graffeo, Danilo Lewański, Andrea T. Ricolfi

TL;DR

This work advances the study of punctual nested Hilbert and Quot schemes on smooth surfaces by combining motivic and topological perspectives. It delivers explicit motivic formulas for two- and three-fold nested Hilbert schemes, links these motives to stratifications by Hilbert–Samuel data, and uses a novel differential-operator framework on Ferrers diagrams to classify Euler-characteristic generating functions. The authors prove that nested Hilbert-characteristic series factor as rational functions with roots-of-unity poles and extend the framework to higher ranks via a master identity connecting nested Hilbert and Quot series, providing concrete evaluations and irreducible-component counts. Overall, the paper provides a coherent toolkit—stratifications, apolarity, and operator methods—that yields both motivic refinements and Euler-characteristic enumerations for nested moduli on surfaces, with clear connections to combinatorial structures and the Grothendieck ring of varieties. The results have potential implications for moduli of flags of sheaves and for DT-type enumerative problems via a unified, motivic-plus-combinatorial approach.

Abstract

Let $(S,p)$ be a smooth pointed surface. In the first part of this paper we study motivic invariants of punctual nested Hilbert schemes attached to $(S,p)$ using the Hilbert-Samuel stratification. We compute two infinite families of motivic classes of punctual nested Hilbert schemes, corresponding to nestings of the form $(2,n)$ and $(3,n)$. As a consequence, we are able to give a lower bound for the number of irreducible components of $S_p^{[2,n]}$ and $S_p^{[3,n]}$. In the second part of this paper we characterise completely the generating series of Euler characteristics of all nested Hilbert and Quot schemes. This is achieved via a novel technique, involving differential operators modelled on the enumerative problem, which we introduce. From this analysis, we deduce that in the Hilbert scheme case the generating series is the product of a rational function by the celebrated Euler's product formula counting integer partitions. In higher rank, we derive functional equations relating the nested Quot scheme generating series to the rank one series, corresponding to nested Hilbert schemes.

Invariants of nested Hilbert and Quot schemes on surfaces

TL;DR

This work advances the study of punctual nested Hilbert and Quot schemes on smooth surfaces by combining motivic and topological perspectives. It delivers explicit motivic formulas for two- and three-fold nested Hilbert schemes, links these motives to stratifications by Hilbert–Samuel data, and uses a novel differential-operator framework on Ferrers diagrams to classify Euler-characteristic generating functions. The authors prove that nested Hilbert-characteristic series factor as rational functions with roots-of-unity poles and extend the framework to higher ranks via a master identity connecting nested Hilbert and Quot series, providing concrete evaluations and irreducible-component counts. Overall, the paper provides a coherent toolkit—stratifications, apolarity, and operator methods—that yields both motivic refinements and Euler-characteristic enumerations for nested moduli on surfaces, with clear connections to combinatorial structures and the Grothendieck ring of varieties. The results have potential implications for moduli of flags of sheaves and for DT-type enumerative problems via a unified, motivic-plus-combinatorial approach.

Abstract

Let be a smooth pointed surface. In the first part of this paper we study motivic invariants of punctual nested Hilbert schemes attached to using the Hilbert-Samuel stratification. We compute two infinite families of motivic classes of punctual nested Hilbert schemes, corresponding to nestings of the form and . As a consequence, we are able to give a lower bound for the number of irreducible components of and . In the second part of this paper we characterise completely the generating series of Euler characteristics of all nested Hilbert and Quot schemes. This is achieved via a novel technique, involving differential operators modelled on the enumerative problem, which we introduce. From this analysis, we deduce that in the Hilbert scheme case the generating series is the product of a rational function by the celebrated Euler's product formula counting integer partitions. In higher rank, we derive functional equations relating the nested Quot scheme generating series to the rank one series, corresponding to nested Hilbert schemes.

Paper Structure

This paper contains 41 sections, 43 theorems, 258 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Euler characteristic results
  3. Nested Hilbert schemes
  4. Nested Quot schemes
  5. Geometry of nested Hilbert schemes and related works
  6. Conventions
  7. Background material
  8. Nested Hilbert schemes
  9. Punctual locus
  10. Curvilinear locus
  11. Tangent space
  12. The Białynicki-Birula decomposition of the punctual nested Hilbert scheme
  13. Tools from commutative algebra
  14. Hilbert--Samuel functions
  15. Hilbert--Samuel strata and homogeneous loci
  16. ...and 26 more sections

Key Result

Theorem A

There are identities where $\mathsf h(t) \in {\mathbb{Z}}[{\mathbb{L}},t] \subset K_0(\mathop{\mathrm{Var}}\nolimits_{{\mathbb{C}}})[t]$ is the polynomial

Figures (7)

  • Figure 1: On the left the Ferrers diagram of a partition $\lambda$. In the center its decomposition in blocks corresponding to each of nonzero multiplicities $m_i(\lambda)$, and marked all possible delimiters between blocks, including the two extremes top and bottom. On the right all possible insertions of a single box in $\lambda$: they correspond one-to-one to blocks delimiters, i.e. to nonzero multiplicities $m_i(\lambda)$, plus one.
  • Figure 2: The north-west path (thick) and the south-east path (thin) of a skew Ferrers diagram $\lambda$.
  • Figure 3: Vertical and horizontal lengths on the north-west path of a skew Ferrers diagram $\lambda$.
  • Figure 4: Let $\rho = (5^{4} 4^0 3^0 2^0 1^{4})$ and let $\lambda$ (for simplicity also a partition $\lambda = (3^1 2^0 1^1)$) with $\ell_1 = 3$ and $v_1 = 2$. The first $\ell_1 = 3$ steps west are described for some $j$ by the operator $\frac{\mathrm{d}}{\mathrm{d}y_{j-2,0}}\frac{\mathrm{d}}{\mathrm{d}y_{j-1,0}}$, which selects the partition $\rho$ for the two values of $j=5$ — since $m_{4}(\rho) = m_{3}(\rho) = 0$ — and of $j=4$ — since $m_{3}(\rho) = m_{2}(\rho) = 0$. The summand for $j=5$ will not survive the application of differential operator for the vertical part, but the $j=4$ summand will. Notice that the $j=4$ summand requires $m_{3}(\rho) = m_{2}(\rho) = 0$ but the fact that $m_4(\rho)$ additionally vanishes does not spoil the application of the operator.
  • Figure 5: Here we see that $\lambda$ is such that $v_1=4$, and that $v_1$ is not the last vertical length of the skew Ferrers diagram (i.e. $M >1$). Therefore one of the conditions for $\rho$ is that the south-east path of $\rho$ needs to have exactly $4$ south steps at some point. If $v_1$ had been the last vertical length of $\lambda$, then $\rho$ could have had at least$4$ steps south along the boundary matching $v_1$ and still be compatible. In order to have $4$ south steps, $\rho$ must have a multiplicity equal to $4$, in this case $m_7(\rho) = 4$. This corresponds to the operator $\frac{\mathrm{d}}{\mathrm{d}y_{7,4}} - \frac{\mathrm{d}}{\mathrm{d}y_{7,5}}$.
  • ...and 2 more figures

Theorems & Definitions (116)

  • Theorem A: Corollaries \ref{['cor:k=2']}, \ref{['cor:k=3']}
  • Theorem B: \ref{['thm:ZDnested']}
  • Theorem C: \ref{['thm:Z_Dr']}, \ref{['cor:FQ/Z^r']}
  • Theorem D: \ref{['thm:Z_Dr3']}
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: ELEMENTARY
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 106 more