How to compute the canonical module of a set of points

Artin Level Modules

“…They were found by F. O. Schreyer. The four first examples were known by Beck and Kreuzer [2] but the last five examples are new.…”

“…Beck and Kreuzer [2] have found a way to compute a presentation of the canonical module ω A of the coordinate ring A of a set of points using Gröbner basis techniques. By this method they have been able to find the four first counterexamples in Table I. …”

Exterior algebra methods for the minimal resolution conjecture

“…The minimal resolution conjecture for a general set of γ points in P r has received considerable attention; see Eisenbud and Popescu [1999] for full references and discussion. In particular, it is known that the minimal resolution conjecture is satisfied if r ≤ 4 (Gaeta [1951] and [1995], Geramita-Lorenzini [1989], Ballico-Geramita [1986], Walter [1995], Lauze [1996]), or γ ≫ r (Hirschowitz-Simpson [1996]), but computer evidence produced by Schreyer, extended also by Mats Boij in his 1994 thesis and by Beck and Kreuzer in [1996] suggested that it might fail for certain examples starting with 11 points in P 6 . Indeed, Eisenbud and Popescu [1999] proved that if r ≥ 6, r = 9, then the minimal resolution conjecture fails for a general set of γ = r + ⌊(3 + √ 8r + 1)/2⌋ points in P r .…”