Generic $μ$-power ideals, fat points, and symbolic powers

Abstract

Problem F of \cite{FLOS18} asks whether an ideal generated by $r$ generic $μ$-power forms, where $μ\vdash d$ is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\big[(1-t^d)^r(1-t)^{-n}\big]_+$ as an ideal generated by $r$ generic forms of degree $d$. We show that the answer is negative, and we determine exactly when the natural obstruction is available. Our main tool is an apolarity inclusion that converts a distinguished large part of $μ$ into a fat-point linear system in the dual projective space. This yields a single \emph{master obstruction} (Theorem \ref{thm:master}): a counterexample exists whenever, for $r$ general points of $\PP^{n-1}$, the homogeneous system $\LL(q;m^r)$ is non-empty while $\LL(q;(m-a)^r)$ has non-positive virtual dimension. We complement this with a positive result (Theorem \ref{thm:small-r}): by Stanley's theorem, Problem F has a positive answer for \emph{every} $μ$ as soon as $r\le n+1$. For ternary forms, the obstruction produces counterexamples precisely for $5\le r\le 8$, with extremal curves given by the conic through five points and products of $(-1)$-curves on del Pezzo surfaces. It is vacuous unconditionally for $r\le 4$ and $r=9$, and, assuming Nagata's conjecture, for every non-pure $μ$ when $r\ge 10$. We also obtain counterexamples in four variables for $r=9$. An exact finite-field certificate shows that $d=14$, $μ=(13,1)$ is the \emph{first} failure for five ternary generators.