Let C⊆Prk be a non-degenerate projective curve of degree r+2, where r≥3. By means of the Hartshorne-Rao module of C we distinguish 4 different possible cases (and an exceptional case which only appears if r=3). In any case C is obtained either by means of an embedding of an arbitrary smooth curve C0 of genus 2, or by projecting an elliptic normal curve from a point or by projecting a rational normal curve from a line. Finally, the possible minimal free resolutions of the homogeneous coordinate ring of C are studied. © 2001 Academic Press.