We consider the Hilbert scheme $H(d,g)$ of space curves $C$ with homogeneous ideal $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ and Rao module $M:=H_{*}^1\left({ℐ}_C\right)$. By taking suitable generizations (deformations to a more general curve) $C^{\prime }$ of $C$, we simplify the minimal free resolution of $I\left(C\right)$ by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of $I\left(C^{\prime }\right)$. Using this for Buchsbaum curves of diameter one ($M_v\ne 0$ for only one $v$), we establish a one-to-one correspondence between the set $𝒮$ of irreducible components of $H(d,g)$ that contain $\left(C\right)$ and a set of minimal...

This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$, whose main results deal with curves which satisfy ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. of diameter, $\mathrm{diam}M\le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\mathrm{H}(d,g)$, at $\left(C\right)$ under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of $C$ turns out to be equivalent to the...