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