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arXiv · 2408.07593

On the Stable Birationality of Hilbert schemes of points on surfaces

Morena Porzio
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Abstract

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.

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Morena Porzio. 2024-09-30. On the Stable Birationality of Hilbert schemes of points on surfaces. https://arxiv.org/abs/2408.07593

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