The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P k . Introduction L...

The purpose of this paper is to establish a Castelnuovo-Mumford regularity bound for threefolds with mild singularities. Let X be non-degenerate normal projective threefold in Pr degree d and codimension e. We prove that if has rational singularities, then reg(X)≤d−e+2. Our very close sharp conjectured by Eisenbud-Goto. When e=2 Cohen-Macaulay Du Bois we obtain the reg(X)≤d−1, also classify ext...