Geometry of Syzygies via Poncelet Varieties

Ilardi, Giovanna ; Supino, Paola ; Vallès, Jean

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 579-589

Résumé

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_{n} = H^{0} (\mathbb{P}^{1}, \mathcal{O}_{\mathbb{P}_{1}} (n))$. We define $\mathfrak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\mathbb{P}^{1}$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute 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.