Lisa Nicklasson: On the Fröberg conjecture and the Lefschetz properties of graded algebras

Abstract:
In 1985 Fröberg gave a conjecture on the Hilbert series of ideals of polynomial rings generated by generic forms. The conjecture, though widely believed to be true, has only been proved in a few cases. In Paper I of this thesis we conjecture, based on computer computations, that ideals generated by powers of generic forms of degree at least two give the same Hilbert series as generic forms.
The topic of Paper II and III is, the so called, Lefschetz properties. An algebra has the strong Lefschetz property if multiplication by any power of a generic linear form has maximal rank. This can also be described via Hilbert series, which connects it to the Fröberg conjecture. The main result of Paper II and Paper III is a classification of monomial complete intersections of positive characteristic, which has the strong Lefschetz property.