On the curves of contact on surfaces in a projective space. III

Abstract:

Suppose a smooth curve $C$ is a set-theoretic complete intersection of two surfaces $F$ and $G$ with the multiplicity of $F$ along $C$ less than or equal to the multiplicity of $G$ along $C$. One obtains a relation between the degrees of $C$, $F$ and $G$, the genus of $C$, and the multiplicity of $F$ along $C$ in case $F$ has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in $\mathbf {P}^3$ is not set-theoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree $\geq 5$.