On the Hilbert series of ideals generated by general linear forms

We determine the Hilbert series of ideals generated by $d$'th powers of $n+2$ general linear forms in $n$ variables, to give upper bounds on the degree of the Hilbert series of ideals generated by $d$'th powers of $n+k$ general linear forms for $k>2$. This allows us to show that the Iarrobino-Fröberg Conjecture fails for all $n$ large enough. We also determine the degree of the Hilbert series for the ideal generated by $d$'th powers of $n+3$ general linear forms, for some values of $n$, and give counterexamples to a conjecture on the failure of the Weak Lefschetz Property for ideals generated by sufficiently large powers of general linear forms. Moreover, we determine the Hilbert series of the ideal generated by two generic quadratic forms in the exterior algebra on an even number of generators.