On the syzygies of quasi-complete intersection space curves

Abstract

Abstract. In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if X is a quasi-complete intersection curve in P 3 , then I X has a minimal free resolutionTherefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of I X . Also we give a relation for the degrees of syzygy modules of I X . Using this theorem, one can construct a smooth quasicomplete intersection curve X such that the number of minimal generators of I X is t for any given positive integer t ∈ Z + . PreliminariesLet X be a nondegenerate locally Cohen-Macaulay irreducible curve in P 3 , defined over an algebraically closed field K of characteristic 0. A curve X ⊂ P 3 is said to be a monomial curve if it has a parametric representation of the form (t n 3 0 , t3 ] be the homogeneous ideal of a monomial curve in P 3 and let µ(I X ) be the number of elements in a minimal generating set of I X . In his paper ([2]), Bresinsky proves that if µ(I X ) = µ ≥ 3, then I X has the following minimal free resolution:where d i,j ∈ Z. Therefore the ranks of the first and the second syzygy modules of the homogeneous ideals of monomial curves in P 3 are determined by the number of elements in a minimal generating set of I X . In this paper, we give a similar result for quasi-complete intersection space curves. Here, a curve X ⊂ P 3 is a