M. Boratynski, ON THE CURVES OF CONTACT ON SURFACES IN A PROJECTIVE-SPACE .3., Proceedings of the American Mathematical Society, 125(2), 1997, pp. 329-338
Suppose a smooth curve C is a set-theoretic complete intersection of t
wo surfaces F and G with the multiplicity of F along C less than or eq
ual to the multiplicity of G along C, One obtains a relation between t
he degrees of C, F and G, the genus of C, and the multiplicity of F al
ong C in case F has only ordinary singularities. One obtains (in the c
haracteristic zero case) that a nonsingular rational curve of degree 4
in P-3 is not set-theoretically an intersection of 2 surfaces, provid
ed one of them has at most ordinary singularities. The same result hol
ds for a general nonsingular rational curve of degree greater than or
equal to 5.