Last syzygies of 1-generic spaces

Abstract

We consider determinantal varieties X ( γ ) of expected codimension defined by the maximal minors of a matrix M ( γ ) of linear forms representing a linear map γ . Eisenbud and Popescu have conjectured that 1-generic linear maps γ have the property that the syzygy ideals I ( s ) of all last syzygies s of X ( γ ) coincide with I X ( γ ) . We prove a geometric version of this conjecture: for 1-generic linear maps γ the syzygy varieties Syz( s )= V ( I ( s )) of all last syzygies have the same support as X ( γ ).