Cohomology of the Quot scheme of an infinite affine space
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915616394313728 |
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| author | Pielasa, Paweł |
| author_facet | Pielasa, Paweł |
| contents | We study the Quot scheme of points $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$. We exhibit and compute the cohomology of explicit loci in $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$, whose complement has codimension diverging to infinity as $n\rightarrow \infty$. In the case $1<r<\frac{d+1}{2}$ this loci is an irreducible component. The main ingredient in our proof are classification results on maximal-dimensional spaces of commutative matrices satisfying certain generating conditions. Our primary motivation is the study of the ind-scheme \[ \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{\infty}}^{\oplus r}) := \underset{n\rightarrow \infty}{colim} \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r}). \] Finally, we compute the cohomology (with integral coefficients) of the Quot scheme $\mathrm{Quot}_2(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})$, confirming, in the case $d=2$, a conjecture of Pandharipande. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomology of the Quot scheme of an infinite affine space Pielasa, Paweł Algebraic Geometry Commutative Algebra 14D23, 14C05, 15A27, 13E10 We study the Quot scheme of points $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$. We exhibit and compute the cohomology of explicit loci in $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$, whose complement has codimension diverging to infinity as $n\rightarrow \infty$. In the case $1<r<\frac{d+1}{2}$ this loci is an irreducible component. The main ingredient in our proof are classification results on maximal-dimensional spaces of commutative matrices satisfying certain generating conditions. Our primary motivation is the study of the ind-scheme \[ \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{\infty}}^{\oplus r}) := \underset{n\rightarrow \infty}{colim} \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r}). \] Finally, we compute the cohomology (with integral coefficients) of the Quot scheme $\mathrm{Quot}_2(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})$, confirming, in the case $d=2$, a conjecture of Pandharipande. |
| title | Cohomology of the Quot scheme of an infinite affine space |
| topic | Algebraic Geometry Commutative Algebra 14D23, 14C05, 15A27, 13E10 |
| url | https://arxiv.org/abs/2511.10742 |