Geometry of syzygies via Poncelet varieties

TL;DR

This paper explores the geometric structure of syzygies in linear systems on projective lines, linking algebraic relations to singularities on Poncelet varieties, and computes the dimension of related classifying spaces.

Contribution

It introduces a new classifying space for linear systems with fixed syzygies and establishes a connection between syzygies and singularities on Poncelet varieties.

Findings

Dimension of the classifying space $rak{X}_{k,r,d}$ computed.

Existence of linear syzygies implies singularities on Poncelet varieties.

Links algebraic syzygies to geometric singularities.

Abstract

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.