It is an exercise in Hartshorne to classify nonsingular quartic curves in projective 3-space. I am interested in what happens when we allow singularities. In particular, I am looking for an explanation or source for how to exclude the possibility of a space curve of arithmetic genus 1 and a single node or cusp.



Quartic Space Curves
Singular nondegenerate irreducible degree $4$ curves certainly exist. They can be obtained as complete intersections of two quadric surfaces which are tangent at some point.


Thinking differently, any curve of class $(2,2)$ on a nonsingular quadric $Q$ is a degree $4$ space curve. The series $|\mathcal{O}\_Q(2)|$ is $8$-dimensional, and the singular curves in this family form a $7$-dimensional subvariety. Cuspidal members of this family form a smaller $6$-dimensional locus.


