  We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear
subspaces of $V_n=H^0({\P^1},\O_{\P^1}(n))$. We define $\frak{X}_{k,r,d}$ 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 $\frak{X}_{k,r,d}$. In the
second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet
varieties. In particular, we prove that the existence of linear syzygies
implies the existence of singularities on the Poncelet varieties.
