Remarks on degree 4 projective curves

In this paper we characterize the degree 4 multiple lines with generic embedding dimension 3 and among them the ones with very degenerate hyperplanc section, and the ones which contain a degree 3 planar subcurve. Using that characterization, we prove that the degree 4 curves containing a planar subcurve of degree 3 are the general element of all irreducible component of the Hilbert scheme. Moreover, we show that all the multiple lines we consider belong to the same connected component of the corresponding Hilbert scheme.