Sextic curves with six double points on a conic

Let C6 be a plane sextic curve with 6 double points that are not nodes. It is shown that if they are on a conic C2, then the unique possible case is that all of them are ordinary cusps. From this it follows that C6 is irreducible. Moreover, there is a plane cubic curve C3 such that C6 = C 3 2 + C 3 . Such curves are closely related to both the branch curve of the projection to a plane of the general cubic surface from a point outside it and canonical surfaces in P or P whose desingularizations have birational invariants q > 0, pg = 4 or pg = 5, P2 ≤ 23.