The syzygy scheme is the scheme defined by the quadric forms associated to the linear syzygies of certain order of a given scheme. It is natural to ask whether the syzygy scheme is equal to the scheme itself. In this talk, I will discuss about the classification of the second syzygy schemes for 4-gonal canonical curves of genus at least 6. This talk is based on the work by Aprodu-Bruno-Sernesi.

In recent years, syzygies of projections of algebraic varieties have drawn a lot of attentions. It turns out that their Betti diagrams carry geometric information like the codimension of the projection and the position of the projection center, by the investigations of E. Park, S. Kwak and so on. In this talk, I will show that for a generic canonical curve $C$ in $\mathbb{P}^{g−1}$, its projection $C'$ away from a generic point into $\mathbb{P}^{g−2}$ is cut out by quadrics for $g \geq 9$. I will also give the predictions of the Betti diagrams with the help of Macaulay2.