Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07593 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917791625379840 |
|---|---|
| author | Porzio, Morena |
| author_facet | Porzio, Morena |
| contents | The aim of this paper is to study the stable birational type of $Hilb^n_X$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n')$ the variety $Hilb^n_X$ is stably birational to $Hilb^{n'}_X$, when $X$ is a surface with irregularity $q(X)=0$. After general results for such surfaces, we restrict our attention to geometrically rational surfaces, proving that there are only finitely many stable birational classes among the $Hilb^n_X$'s. As a corollary, we deduce the rationality of the motivic zeta function $ζ(X,t)$ in $K_0(Var/k)/([\mathbb{A}^1_k])[[t]]$ over fields of characteristic zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07593 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Stable Birationality of Hilbert schemes of points on surfaces Porzio, Morena Algebraic Geometry 14E08, 14G10, 14M20, 14C05 The aim of this paper is to study the stable birational type of $Hilb^n_X$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n')$ the variety $Hilb^n_X$ is stably birational to $Hilb^{n'}_X$, when $X$ is a surface with irregularity $q(X)=0$. After general results for such surfaces, we restrict our attention to geometrically rational surfaces, proving that there are only finitely many stable birational classes among the $Hilb^n_X$'s. As a corollary, we deduce the rationality of the motivic zeta function $ζ(X,t)$ in $K_0(Var/k)/([\mathbb{A}^1_k])[[t]]$ over fields of characteristic zero. |
| title | On the Stable Birationality of Hilbert schemes of points on surfaces |
| topic | Algebraic Geometry 14E08, 14G10, 14M20, 14C05 |
| url | https://arxiv.org/abs/2408.07593 |