Eric Dannetun: On invariants of generic ideals

Abstract: Given a parametrized family of ideals in a polynomial ring, many invariants, such as the Hilbert series and Betti numbers, are often constant on a Zariski open subset of the parameter space, and hence it makes sense to speak of these invariants for a generic ideal. This thesis studies these invariants for two types of generic ideals.

Hereafter, the term "generic ideal" will refer to a generic ideal generated by a fixed number of generators of prescribed degrees. In Paper 1, we investigate the Hilbert series of a generic ideal generated by \(r\) forms of degree \(d\). The Fröberg conjecture is a prediction for this series, and we prove that it holds in degree \(d+2\) when \(d>2\). Moreover, we show that the conjecture is true up to degree \(2d-1\), assuming that the number of variables is sufficiently large. In Paper 2, we consider a generic ideal among ideals generated by \(n+1\) powers of linear forms of degrees \(d_1, \dots,d_{(n+1)}\) in a polynomial ring with \(n\) variables. It is known that the Hilbert series of such an ideal equals that of a generic ideal generated in the corresponding degrees, and moreover that this is equal to the series predicted by the Fröberg conjecture. We work towards understanding their Betti numbers, which form a finer invariant, and our main result determines the Betti numbers when at least one generator is a square. We also determine the Betti numbers of the Artinian Gorenstein algebras which are linked to these almost complete intersections, and we give an explicit set of generators for the defining ideals in some cases. Our results provide upper bounds for the Betti numbers of a generic ideal generated by \(n+1\) forms with at least one quadric generator, and we are able to conclude that the corresponding quotient algebra is level.