The Minimal Resolution Conjecture Original Research Article

By extending the ideal generation conjecture, we formulate the minimal resolution conjecture for a sufficiently generic set of points in imagen and we explicitly write out the conjectured values of the Betti numbers. We then relate the minimal resolution conjecture to the ideal generation conjecture and the Cohen-Macaulay type conjecture, and we prove the minimal resolution conjecture for (d + nn) − λ (1 ≤ λ ≤ n) points, for n + 2 points, and for (d − 1 + nn) points. We also prove the Cohen-Macaulay type conjecture for s points, with n + 1 ≤ s ≤ (2 + nn)! Finally, we recover information about the resolution of s − 1 or s + 1 points from that of S points.