  A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional
subscheme of projective 3-space whose homogeneous ideal is the intersection of
powers of the ideals of the six coordinate lines. The second and third authors
have shown that these curves have very nice combinatorial properties, and they
have made a careful study of the even liaison classes of these curves. We build
on this work by showing that they are "almost always" componentwise linear,
i.e. their homogeneous ideals have the property that for any d, the degree d
component of the ideal generates a new ideal whose minimal free resolution is
linear. The one type of exception is clearly spelled out and studied as well.
The main technique is a careful study of the way that basic double linkage
behaves on tetrahedral curves, and the connection to the tetrahedral curves
that are minimal in their even liaison classes. With this preparation, we also
describe the minimal free resolution of a tetrahedral curve, and in particular
we show that in any fixed even liaison class there are only finitely many
tetrahedral curves with linear resolution. Finally, we begin the study of the
generic initial ideal (gin) of a tetrahedral curve. We produce the gin for
arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically
Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal
tetrahedral curve in terms of the gin of the minimal curve in that even liaison
class.
