On the Ideal of Minors of Matrices of Linear Forms

The ideals generated by the minors of matrices whose entries are linear forms are not yet well-understood, unless the forms themselves satisfy some strong condition. One has a wealth of information if the matrix is generic, symmetric generic or Hankel; here we tackle 1-generic matrices. We recall the definition of 1-genericity introduced in [E2] by Eisenbud: Let F be a field and X1, . . . , Xs be indeterminates over F . Let M be an m× n matrix of linear forms in F [X1, . . . , Xs], with m ≤ n and s ≥ m + n− 1. By a generalized row of M one means a non-trivial F -linear combination of the rows of M . By a generalized entry of M one means a non-trivial F -linear combination of the entries of a generalized row of M . M is said to be 1-generic if every generalized entry is non-zero. Generic, symmetric generic, Hankel matrices, as well as many others are all 1-generic. In this wider context, however, the only case of determinantal ideals fully understood is that of the ideals generated by the maximal minors. In fact, in [E2] it is proved that these ideals are prime. However, when one considers ideals generated by non-maximal minors, patterns get complicated by the fact that often these ideals are not prime.