Complete intersections primitive structures on space curves (Q2788772)

For \(C\) a smooth space curve, the author studies numerical conditions for \(C\) to have a primitive multiple structure \(X\) which is a complete intersection of two surfaces \(F_{a}\), \(F_{b}\). A multiple structure \(X\) on \(C\) is a locally Cohen-Macaulay curve whose support is \(C\) and \(\deg(X)=\mathrm{mdeg}(C)\) where \(m\) is the multiplicity of \(X\). \textit{C. Bănică} and \textit{O. Forster} [Contemp. Math. 58, 47--64 (1986; Zbl 0605.14026)]. The author shows that for fixed \(d=\deg(C)\) and \(g=\mathrm{genus}(C)\) there are only finitely many possible \((a,b,m)\) for \(C\) to be a primitive set theoretic complete intersection (see also [\textit{D. B. Jaffe}, J. Reine Angew. Math. 464, 1--45 (1995; Zbl 0826.14021)]). In his main theorem the author gives specific numerical conditions that need to be satisfied for \(C\) to be a primitive set theoretic complete intersection. As a corollary he concludes that for the smooth rational quartic curve there are exactly 10 possible cases (see also [\textit{D. B. Jaffe}, Math. Ann. 294, No. 4, 645--660 (1992; Zbl 0757.14020)]).