Geometry of syzygies via Poncelet varieties

Giovanna Ilardi (1) , Paola Supino (2) , Jean Vallès (3)

1 Dip. Matematica e Applicazioni di Napoli
2 Dipartimento di Matematica [Roma TRE]
3 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\mathbb{P}^1},O_{\mathbb{P}^1}(n))$. 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 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.

Mots clés: Syzygies of binary forms; Poncelet varieties; Grassmanian

Domaines: Géométrie algébrique [math.AG]

Giovanna Ilardi, Paola Supino, Jean Vallès. Geometry of syzygies via Poncelet varieties. Bollettino dell'Unione Matematica Italiana, 2009, 9 (3), pp.579-589. ⟨hal-00292071⟩