Prime Ideals and Three-generated Ideals with Large Regularity

Ananyan and Hochster proved the existence of a function Φ(m,d) such that any graded ideal I generated by m forms of degree at most d in a standard graded polynomial ring satisfies reg(I)≤Φ(m,d). Relatedly, Caviglia et. al. proved the existence of a function Ψ(e) such that any nondegenerate prime ideal P of degree e in a standard graded polynomial ring over an algebraically closed field satisfies reg(P)≤Ψ(deg(P)). We provide a construction showing that both Φ(3,d) and Ψ(e) must be at least doubly exponential in d and e, respectively. Previously known lower bounds were merely super-polynomial in both cases.