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arXiv · 1207.4588

Smooth curves specialize to extremal curves

Robin Hartshorne·Paolo Lella·Enrico Schlesinger
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Abstract

Let $H_{d,g}$ denote the Hilbert scheme of locally Cohen-Macaulay curves of degree $d$ and genus $g$ in projective three space. We show that, given a smooth irreducible curve $C$ of degree $d$ and genus $g$, there is a rational curve $\{[C_t]: t \in \mathbb{A}^1\}$ in $H_{d,g}$ such that $C_t$ for $t \neq 0$ is projectively equivalent to $C$, while the special fibre $C_0$ is an extremal curve. It follows that smooth curves lie in a unique connected component of $H_{d,g}$. We also determine necessary and sufficient conditions for a locally Cohen-Macaulay curve to admit such a specialization to an extremal curve.

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Robin Hartshorne, Paolo Lella, Enrico Schlesinger. 2012-07-19. Smooth curves specialize to extremal curves. https://arxiv.org/abs/1207.4588

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