  The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial
expansion of the symmetric function $\Delta'_{e_{k-1}} e_n$, where $k \leq n$
are positive integers and $\Delta'_{e_{k-1}}$ is a Macdonald eigenoperator.
When $k = n$, the specialization $\Delta'_{e_{n-1}} e_n|_{t = 0}$ is the
Frobenius image of the graded $S_n$-module afforded by the cohomology ring of
the {\em flag variety} consisting of complete flags in $\mathbb{C}^n$. We
define and study a variety $X_{n,k}$ which carries an action of $S_n$ whose
cohomology ring $H^{\bullet}(X_{n,k})$ has Frobenius image given by
$\Delta'_{e_{k-1}} e_n|_{t = 0}$, up to a minor twist. The variety $X_{n,k}$
has a cellular decomposition with cells $C_w$ indexed by length $n$ words $w =
w_1 \dots w_n$ in the alphabet $\{1, 2, \dots, k\}$ in which each letter
appears at least once. When $k = n$, the variety $X_{n,k}$ is homotopy
equivalent to the flag variety. We give a presentation for the cohomology ring
$H^{\bullet}(X_{n,k})$ as a quotient of the polynomial ring $\mathbb{Z}[x_1,
\dots, x_n]$ and describe polynomial representatives for the classes $[
\overline{C}_w]$ of the closures of the cells $C_w$; these representatives
generalize the classical Schubert polynomials.
