The Hilbert Schemes of Degree Three Curves are Connected

In this paper we show that the Hilbert scheme $H(3,g)$ of locally Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected. In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many irreducible components (roughly $-g/3$ of them). To show connectedness, we classify the curves (giving particular attention to the triple lines), determine the irreducible components, and give flat families over $\Aone$ to show that the components meet. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.