MATH Seminar
| Title: On the stable birationality of Hilbert schemes of points on surfaces |
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| Seminar: Algebra and Number Theory |
| Speaker: Morena Porzio of Columbia University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-11-19 at 5:00PM |
| Venue: MSC W303 |
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| Abstract: The Cassels--Swinnerton-Dyer conjecture says that cubic surfaces of index 1 have a rational point. This can be reformulated into a statement about the existence of rational maps between Hilbert schemes of points $\mathrm{Hilb}^n_X$, which in turn motivates the study of the stable birational type of $\mathrm{Hilb}^n_X$. In this talk, we will address the question for which pairs of integers $(n,n’)$ the variety $\mathrm{Hilb}^n_X$ is stably birational to $\mathrm{Hilb}^{n'}_X$, when $X$ is a surface with $H^1(X,\mathcal{O}_X)=0$. In order to do so we will relate the existence of degree $n'$ effective cycles on $X$ with the existence of degree $n$ ones using curves on $X$. We will then focus on geometrically rational surfaces, proving that there are finitely many birational classes among $\mathrm{Hilb}^n_X$'s. |
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