Characterize cases where the Eisenbud–Goto inequality holds

Establish when the Eisenbud–Goto inequality \(\operatorname{reg}(P)\leq \deg(P)-\operatorname{height}(P)+1\) holds for homogeneous prime ideals over an algebraically closed field, despite its failure in general.

Background

The paper introduces the Eisenbud–Goto conjecture, which asserted that the Castelnuovo–Mumford regularity of a homogeneous prime ideal is bounded above by its multiplicity minus its height plus one. Although McCullough and Peeva constructed counterexamples showing that the conjecture is false for arbitrary homogeneous primes, determining classes of varieties or prime ideals for which the inequality remains valid continues to be an unresolved problem.

The paper discusses several positive cases, including arithmetically Cohen–Macaulay varieties, projective curves, and certain classes of surfaces, as well as known counterexamples in higher dimensions and for singular surfaces. The stated open problem is therefore to identify the circumstances under which the Eisenbud–Goto inequality still holds, rather than to prove the conjecture in full generality.