Codice QDD 141 Titolo Smooth curves specialize to extremal curves Data 2012-11-15 Autore/i Hartshorne, R.; Lella, P.; Schlesinger, E: Link Download full text 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]} 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.