Codimension two ACM varieties in P1xP1xP1 and regularity of bicyclic graphs and their powers

In this PhD thesis, we discuss several different results about some homological invariants (e.g., graded Betti numbers, Hilbert function, regularity) of some special varieties. In particular, we focus on the codimension two ACM varieties in P1×P1×P1 (called varieties of lines), and the edge ideals of bicyclic graphs. We study the Hilbert function of Ferrers varieties of lines, a special case of ACM variety of lines, and we describe the trigraded minimal free resolution of the defining ideal of a variety of lines arising from a complete intersection of points. We also compute the Castelnuovo-Mumford regularity of the defining ideal of grids of lines and complete intersections of lines in P1×P1×P1. Then we study the regularity of another special variety, i.e., the edge ideal of a bicyclic graph and its powers. Specifically, we compute the regularity of the edge ideal of a dumbbell graph, and then we give a combinatorial characterization of the regularity of the edge ideal of an arbitrary bicyclic graph in terms of its induced matching number. Finally we study the regularity of powers of edge ideals of some specific bicyclic graphs, i.e., dumbbell graphs with path having at most two vertices