The information encoded in initial ideals

Abstract

We consider homogeneous ideals $I$ and the initial ideal $\text {in}(I)$ for the revlex order. First we give a sequence of invariants computed from $I$ giving better and better “approximations" to the initial ideal and ending in an equivalent description. Then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme. We also consider the higher initial ideals as defined in a paper by Fløystad. In particular, we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve.