Space curves which are the intersection of a cone with another surface (Q1123233)

Let \(S\subseteq {\mathbb{P}}^ 3\) be a cone and let \(C\subseteq S\) be a smooth curve. This paper gives necessary and sufficient conditions for there to exist a surface \(T\subseteq {\mathbb{P}}^ 3\) such that \(C=S\cap T\) as sets. In this case C is called a set-theoretic intersection on S. The conditions are phrased in terms of the interaction between C and a generic hyperplane section D of S. For example, if the base field has characteristic zero and the vertex of S is not on C, then C is a set- theoretic complete intersection on S if and only if D is smooth. One consequence of this characterization, is the proof that, in characteristic zero, a smooth monomial curve of degree at least four cannot be a set-theoretic complete intersection on a cone.