Abstract
We study the set of circuits of a homogeneous ideal and that of its truncations, and introduce the notion of generic circuits set. We show how this is a well-defined in- variant that can be used, in the case of initial ideals with respect to weights as a counterpart of the (usual) generic ini- tial ideal with respect to monomial orders. As an applica- tion we recover the existence of the generic fan introduced by Romer and Schmitz for studying generic tropical varieties. We also consider general initial ideals with respect to weights and show, in analogy to the fact that generic initial ideals are Borel-fixed, that these are fixed under the action of certain Borel subgroups of the general linear group.
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