  We study the geometry and topology of $\Delta$-Springer varieties associated
with two-row partitions. These varieties were introduced in recent work by
Griffin-Levinson-Woo to give a geometric realization of a symmetric function
appearing in the Delta conjecture by Haglund-Remmel-Wilson. We provide an
explicit and combinatorial description of the irreducible components of the
two-row $\Delta$-Springer variety and compare it to the ordinary two-row
Springer fiber as well as Kato's exotic Springer fiber corresponding to a
one-row bipartition. In addition to that, we extend the action of the symmetric
group on the homology of the two-row $\Delta$-Springer variety to an action of
a degenerate affine Hecke algebra and relate this action to a
$\mathfrak{gl}_{2}$-tensor space.
