Groebner bases of the ideal of a space curve (Q1977892)

Let \(C \subseteq {\mathbb P}^3\) be a curve of degree \(d\) and genus \(p\), let \(I\) be the affine ideal of \(C\) in \(k[x,y,z]\) and assume that \(C\) satisfies a suitable hypothesis of general position. The aim of the paper is to describe certain Gröbner bases of \(I\). In particular, it is shown that the reduced Gröbner basis of \(I\) with respect to the pure lex order has four elements and the leading terms of these elements are \(x^d\), \(z^2\), \(zx\) and \(zy^D\) (where \(D:={1 \over 2}(d-1)(d-2)-p\)). Moreover some specific description of these four polynomials can be given. Successively the author considers also the case of the elimination term order on \(k[x,y,z]\) which restricts to graded lex on \(k[x,y]\). Also in this case many information on the corresponding reduced Gröbner basis of the ideal \(I\) can be given: It is possible to determine the cardinality of the basis and several conditions on the involved polynomials.