Abstract: In this paper, we study the question about the syzygies of points in the projective plane. Firstly, let $X$ be a finite set consists of 7 distinct points in projective plane $\mathbb{P}^2$. We give the representation of syzygies of $X$, and the minimal free resolutions of corresponding saturated homogeneous ideal $I_X$. According to the number and position of the base points of the linear system, all planar cubic linear systems are classified, and 11 different planar cubic linear systems are obtained.

Key words: Minimal free resolution; , Homogeneous ideal; , Algebraic curve; , Cubic linear system; , Base point