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A note on Hilbert schemes of nodal curves

Ziv Ran
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Abstract

We study completely the Hilbert scheme and punctual Hilbert scheme of a nodal curve, and the relative Hilbert scheme of a family of curves acquiring a node. The results are then extended, less completely, to flag Hilbert schemes, parametrizing chains of subschemes. We find, notably, that if the total space $X$ of a family $X/B$ is smooth (over an algebraically closed field $\k$), then the relative Hilbert scheme $Hilb_m(X/B)$ is smooth over $\k$ and the flag Hilbert schemes are normal and locally complete intersection, but generally singular.

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Ziv Ran. 2004-10-03. A note on Hilbert schemes of nodal curves. https://arxiv.org/abs/math/0410037

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