An algorithm for computing minimal curves
Let \(\mathbb{P}^3\) be the 3-dimensional projective space over an algebraically closed field \(K\) of characteristic 0, and \(R\) be the graded polynomial ring \(K[x_0,x_1,x_2,x_3]\). A curve \(C\) of \(\mathbb{P}^3\) will be a closed subscheme of \(\mathbb{P}^3\), purely of dimension 1, locally Cohen-Macaulay, that is without embedded points, defined by an ideal sheaf \({\mathcal I}_C\). Two curves \(C\) and \(C'\) of \(\mathbb{P}^3\) are said to be geometrically linked if \(C\) and \(C'\) do not have common irreducible components and their union is a complete intersection. The equivalence relation generated by this relation is called liaison. Hence \(C\) and \(C'\) belong to the same liaison class if there exist curves \(C_0= C,C_1,\dots, C_n= C'\) such that \(C_1\) and \(C_{i+1}\) are geometrically linked, for \(i=0,\dots, n-1\). If in particular the integer \(n\) is even, then we say that \(C\) and \(C'\) belong to the same biliaison class. Let \(M_C:= \bigoplus_{n\in\mathbb{Z}} H^1{\mathcal I}_C(n)\) be the Hartshorne-Rao module of the curve \(C\): it is a graded \(R\)-module of finite length which, up to a shift in degrees, characterizes the biliaison class of \(C\). \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)] have proved that in every biliaison class of non arithmetically Cohen-Macaulay curves there exists a minimal curve, unique up to a deformation with constant cohomology and Hartshorne-Rao module, that is a curve \(C\) such that, if \(C_1\) is any curve in the biliaison class of \(C\), then \(M_{C_1}= M_C(-n)\), with \(n\geq 0\). They have moreover given an algorithm for the computation of a minimal curve, based on the computation and the analysis of the minors of given orders of some submatrices of the second syzygy matrix \(\sigma_2\) of \(M\). The aim of this paper is to give an improvement to the algorithm for computing a minimal curve of a given Hartshorne-Rao module. The key remark is that the information obtained from \(\sigma_2\) can analogously be obtained from the matrix \(f(\sigma_2)\) with entries in the polynomial ring \(K[u]\), where \(f:R\to K[u]\) is a map which evaluates the variates of \(R\) to random linear polynomials in \(K[u]\). The advantage of this approach is first, that the computations are done in a simpler polynomial ring and second, that in \(K[u]\) we can take advantage of the Smith normal form algorithm to analyze the structure of the minors of the given matrices. The algorithm is therefore probabilistic but rather efficient and allows to compute (the variants of) a minimal curve associated to modules whose syzygies matrices are considerably large. The algorithms presented here have been partially implemented and tested in the computer algebra systems CoCoA, Maple and Macaulay.
scientific article; zbMATH DE number 1028197