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Smooth curves specialize to extremal curves

Research paper by Robin Hartshorne, Paolo Lella, Enrico Schlesinger

Indexed on: 03 Aug '14Published on: 03 Aug '14Published in: Mathematische Annalen



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 \(t \ne 0\) 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.