Feigin-Odesskii brackets, syzygies, and Cremona transformations

Abstract

We identify Feigin-Odesskii brackets qn,1(C), associated with a normal elliptic curve of degree n, C⊂Pn−1, with the skew-symmetric n×n matrix of quadratic forms introduced by Fisher in [6] 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 Pn−1, generalizing the quadro-cubic Cremona transformation of P4. We identify this transformation with the one considered in [14] and find explicit formulas for the inverse transformation. We also find polynomial formulas for Cremona transformations from [14] associated with higher rank bundles on C.