Geodesic
Geometry of Syzygies via Poncelet Varieties
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 579-589.

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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. 
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Ilardi, Giovanna; Supino, Paola; Vallès, Jean. Geometry of Syzygies via Poncelet Varieties. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 579-589. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_3_a3/

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