On the syzygies of quasi-complete intersection space curves

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 $\mathbb{P}^3$, then $I_X$ has a minimal free resolution

$$0\to \oplus_{i=1}^{\mu-3} S(d_{i+3}+c_1)\to \oplus_{i=1}^{2\mu-4}S(-e_i)\to \oplus_{i=1}^\mu S(-d_i)\to I_X\to 0,$$

where $d_i,e_i\in \mathbb{Z}$ and $c_1=-d_1-d_2-d_3$. Therefore 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 quasi-complete intersection curve $X$ such that the number of minimal generators of $I_X$ is $t$ for any given positive integer $t\in\mathbb{Z}^+$.