Counterexamples to the Eisenbud-goto Regularity Conjecture

Abstract. Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan-Hochster.