The equations of space curves on a quadric (Q2642845)

Here the authors give explicit equations for any curve (i.e. pure one-dimensional locally Cohen-Macaulay subscheme) on any quadric surface \(Q\). When \(Q\) is a double plane, this was done by \textit{N. Chiarli}, \textit{S. Greco} and \textit{U. Nagel} [J. Pure Appl. Algebra 190, No. 1--3, 45--57 (2004; Zbl 1064.14029)], which generalizes [\textit{R. Harshorne} and \textit{E. Schlesinger}, Commun. Algebra 28, No. 12, 5655--5676 (2000; Zbl 0994.14003)]. This is a very natural problem, because such curves arise quite often (e.g. as the ones with extremal properties) and giving ``equations'' means also giving ``parameter spaces''. As an application they give all Hartshorne-Rao modules of space curves lying on a quadric surface.