  Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let
$\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle
of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a
resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in
$X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system
of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$
set-theoretically. This also yields a system of defining equations for the
corresponding orbital varieties $\mu(T^*_XX_w)$. Inspired by the system of
defining equations, we conjecture a type-independent equality, namely
$T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))$. The set-theoretic version
of this conjecture follows from this work and previous work for any cominuscule
Grassmannian of type A, B, or C.
