Springer fibers and the Delta Conjecture at $t=0$

We introduce a family of varieties $Y_{n,lambda,s}$, which we call the $Delta$-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,lambda,s})$ and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induced Specht modules. The $lambda=(1^k)$ case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at $t=0$. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety $Y_{n,lambda}$ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud-Saltman rank variety and diagonal matrices.