Explicit Free Resolutions of Tangent Cones in Affine 4-Space

This paper is devoted to the study of minimal free resolutions of tangent cones associated with complete intersection monomial curves in affine 4-space. By applying the gluing technique for numerical semigroups, we characterize specific families of these curves where the tangent cone retains the complete intersection property. Our analysis focuses particularly on curves defined by 4-generated numerical semigroups, constructed by gluing two semigroups generated by two elements. Furthermore, we present explicit computations of their minimal free resolutions under specific conditions.