Counterexamples to the Eisenbud–Goto regularity conjecture

Abstract

Our main theorem shows that the regularity of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k 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 I , produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I I . The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan and Hochster.