Lower Bounds for the Number of Generic Initial Ideals

الأصل · EN

Given a graded ideal I in a polynomial ring over a field K it is well known, that the number of distinct generic initial ideals of I is finite. While it is known that for a given d∈ there is a global upper bound for the number of generic initial ideals of ideals generated in degree less than d, it is not clear how this bound has to grow with d. In this note we will explicitly give a family (I(d))d∈ of ideals in S=K[x,y,z], such that I(d) is generated in degree d and the number of generic initial ideals of I(d) is bounded from below by a linear bound in d. Moreover, this bound holds for all graded ideals in S, which are generic in an appropriate sense.