Regularity jumps for powers of ideals

The Castelnuovo-Mumford regularity $\text{reg}(I)$ is one of the most important invariants of a homogeneous ideal $I$ in a polynomial ring. A basic question is how the regularity behaves with respect to taking powers of ideals. It is known that in the long-run $\text{reg}(I^k)$ is a linear function of $k$. We show that in the short-run the regularity of $I^k$ can be quite “irregular”. For any given integer $d>1$ we construct an ideal $J$ generated by $d+5$ monomials of degree $d+1$ in 4 variables such that $\text{reg}(J^k)=k(d+1)$ for every $k<d$ and $\text{reg}(J^d)\ge d(d+1)+d-1$.