Computing minimal generators of ideals of elliptic curves (Q2754412)

A curve \(C \subset {\mathbb P}^n\) has ``maximal rank'' if the natural restriction map \(\rho_C(t): H^0({\mathbb P}^n, {\mathcal O}_{{\mathbb P}^n} (t)) \rightarrow H^0(C, {\mathcal O}_C (t))\) has maximal rank, for every integer \(t\). \textit{F. Orecchia} [J. Pure Appl. Algebra 155, 77-89 (2001; Zbl 1032.14015)] gave a characterization of the maximal rank property for disjoint unions of rational curves in terms of the Hilbert function. He also gave results on the number of minimal generators of the ideal of such a union of rational curves. This paper generalizes those results to disjoint unions of smooth non special curves. (\(C\) is non special if \(\max \{ t H^1 ({\mathcal O}_C(t)) \neq 0 \} = 0\).) Much of this work involves a careful study of the Castelnuovo-Mumford regularity of such curves. NEWLINENEWLINENEWLINEThe paper also gives an effective method of computing an elliptic curve of any degree \(d \geq 3\) via its rational points. The authors assume that they work over a field of characteristic \(p \geq 3\). They have an implementation of this construction that is available online. NEWLINENEWLINENEWLINEFinally, the two parts of this paper are merged by a study of the maximal rank and the minimal generation properties for disjoint unions of degree \(d \leq 30\) of \(s \leq 6\) distinct elliptic curves in \({\mathbb P}^n\), where \(3 \leq n \leq 30\). The authors also show the existence of smooth elliptic quintic curves on a general threefold in \({\mathbb P}^4\). They end with a remark that their construction of elliptic curves is more difficult to apply over the rational field \(\mathbb Q\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00013].