Ordered set partitions, Garsia-Procesi modules, and rank varieties

We introduce a family of ideals $I_{n,\lambda,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ partition $k\leq n$ and an integer $s \geq \ell(\lambda)$. This contains both the Tanisaki $I_\lambda$ $I_{n,k}$ Haglund-Rhoades-Shimozono as special cases. study corresponding quotient rings $R_{n,\lambda,s}$ symmetric group modules. When $n=k$ $s$ is arbitrary, we recover Garsia-Procesi modules, when $\lambda=(1^k)$ $s=k$, generalized coinvariant algebras Haglund-Rhoades-Shimozono. give monomial basis $R_{n,\lambda,s}$, unifying bases studied by Haglund-Rhoades-Shimozono, realize $S_n$-module structure terms action on $(n,\lambda,s)$-ordered set partitions. also prove formulas Hilbert series graded Frobenius characteristic $R_{n,\lambda,s}$. then connect our work with Eisenbud-Saltman rank varieties using results Weyman. As application work, basis, formula, formula coordinate ring scheme-theoretic intersection variety diagonal matrices.