Geometry of syzygies via Poncelet varieties

Abstract

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\P }^1,{Ø}_{{\P }^1}(n))$. We define ${\mathfrak{X}}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on ${\P }^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of ${\mathfrak{X}}_{k,r,d}$. In the second part we make a link between ${\mathfrak{X}}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.