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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.11538 |
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| _version_ | 1866915687973257216 |
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| author | Bérczi, Gergely Minddal, Felix |
| author_facet | Bérczi, Gergely Minddal, Felix |
| contents | We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Virtual invariants from the non-associative Hilbert scheme Bérczi, Gergely Minddal, Felix Algebraic Geometry We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds. |
| title | Virtual invariants from the non-associative Hilbert scheme |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.11538 |