Nested Hilbert Schemes and the nested $q,t$-Catalan series
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Bibliographic record
Abstract
In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series. Dans cet article, nous étudions les espaces tangents du schéma de Hilbert emboité lisse $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ de points du plan, et donnons une formule générale pour le calcul de la caractéristique d’Euler d’un faisceau $\mathbb{T}^2$-équivariant localement libre sur $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. En appliquant notre resultat a un faisceau particulier, nous conjecturons que le résultat est un polynôme en$q$ et $t$ à coefficents positifs ou nuls. Nous appelons ce polynôme conjecturalement positif la “série de $q; t$-Catalan emboîtée”, car il a de nombreuses propriétés (conjecturées) similaires à celles de la série de $q; t$-Catalan.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.002 | 0.045 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it