Complete intersection monomial curves and free resolutions

Abstract

In this article, we compute the minimal free resolutions of the tangent cones of complete intersection monomial curves in affine 4-space under certain conditions. We first show that a complete intersection monomial curve in 3-space has a complete intersection tangent cone if and only if it is obtained by a nice gluing. By using the technique of nice gluing, we construct infinitely many families of complete intersection monomial curves with complete intersection tangent cones. We also construct infinitely many families of the complete intersection monomial curves that do not have complete intersection tangent cones and compute the minimal free resolutions of these families.