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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan and Hochster.