Feigin-Odesskii brackets, syzygies, and Cremona transformations

TL;DR

This paper links Feigin-Odesskii brackets with skew-symmetric matrices, explores Cremona transformations derived from secant varieties of elliptic curves, and provides explicit formulas for these transformations and their inverses.

Contribution

It identifies Feigin-Odesskii brackets with Fisher's matrices, and generalizes Cremona transformations related to secant varieties of elliptic curves.

Findings

Identified Feigin-Odesskii brackets with Fisher's matrices.

Established Cremona transformations from secant varieties for odd n.

Derived explicit formulas for transformations and their inverses.

Abstract

We identify Feigin-Odesskii brackets $q_{n,1}(C)$, associated with a normal elliptic curve of degree $n$, $C\subset {\mathbb P}^{n-1}$, with the skew-symmetric $n\times n$ matrix of quadratic forms introduced by Fisher in arXiv:1510.04327 in connection with some minimal free resolutions related to the secant varieties of $C$. On the other hand, we show that for odd $n$, the generators of the ideal of the secant variety of $C$ of codimension $3$ give a Cremona transformation of ${\mathbb P}^{n-1}$, generalizing the quadro-cubic Cremona transformation of ${\mathbb P}^4$. We identify this transformation with the one considered in arXiv:alg-geom/9712022 and find explict formulas for the inverse transformation. We also find polynomial formulas for Cremona transformations from arXiv:alg-geom/9712022 associated with higher rank bundles on $C$.