We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of reg I(G) and the asymptotic linear function reg I(G) q , for q ≥ 1, in terms of combinatorial data of the given graph G.

In this paper, we study arithmetic Macaulayfication of projective schemes and Rees algebras of ideals. We discuss the existence of an arithmetic Macaulayfication for projective schemes. We give a simple neccesary and sufficient condition for nonsingular projective varieties to possess an arithmetic Macaulayfication (Theorem 1.5). We also show that this condition is sufficient in general, but give examples to show that it is not in general necessary. We further consider Rees algebras R (I ) = R[I t] (truncated Rees algebras) associated to a homogeneous ideal I and show that they are Cohen-Macaulay for large in some important cases (Theorem 2.1 and Corollary 2.2.1).

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