Lower Bounds for the Number of Generic Initial Ideals

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 $d\in\N$ 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 $(I(d))_{d\in\N}$ 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.