The action of $GL\_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive dimensonal fibers over the 14 and 9 dimensional orbits (probably some flag variety?).



springer resolution over $\wedge^3 \mathbb{C}^6$
A representative of the quartic orbit is $e\_{156} + e\_{246} + e\_{345}$ (I write $e\_{ijk}$ for $e\_i\wedge e\_j\wedge e\_k$). A resolution of this orbit is given by the projective closure of the tangent bundle to the Grassmannian. Indeed, if $V = \mathbb{C}^6$ and $U$ is the tautological bundle on $X = Gr(3,V)$ then 
$$
T\_X \cong U^\*\otimes V/U \cong \Lambda^2U\otimes V/U\otimes O(1),
$$
so $T\_X(-1) \cong \Lambda^2U \otimes V/U$. Note that the tautological filtration $U\subset V$ induces a filtration of $\Lambda^3 V$ with factors $\Lambda^3U$, $\Lambda^2U\otimes V/U$, $U\otimes\Lambda^2(V/U)$, and $\Lambda^3(V/U)$. Consequently, there is a canonical exact sequence
$$
0 \to T^+ \to \Lambda^3 V\otimes O\_X \to T^- \to 0,
$$
where $T^+$ is the extension of $\Lambda^2U\otimes V/U$ by $\Lambda^3U$ and $T^-$ is the extension of $\Lambda^3(V/U)$ by $U\otimes\Lambda^2(V/U)$. In particular, $T^+$ fits into exact sequence
$$
0 \to O\_X(-1) \to T^+ \to T\_X(-1) \to 0.
$$
The embedding $T^+\to \Lambda^3 V\otimes O\_X$ induces a map $f:P\_X(T^+) \to P(\Lambda^3 V)$. Its image is the invariant quartic hypersurface. 


EDIT: Concerning the fibers of the map $f$. It is easy to show that the fiber of $f$ over a point $\lambda \in P(\Lambda^3V)$ is the subvariety of $Gr(3,V)$ consisting of those $U \subset V$ such that 
$$
\lambda \wedge \Lambda^2U = 0.
$$
Using this it is easy to describe the fibers over representatives of the orbits:
$$
f^{-1}(e\_{156}+e\_{246}+e\_{345}) = \langle e\_4,e\_5,e\_6 \rangle \in Gr(3,V),
$$
$$
f^{-1}(e\_{123}+e\_{145}) = \{ U\ |\ e\_1 \in U \subset \langle e\_1,e\_2,e\_3,e\_4,e\_5 \rangle,\ (e\_{23}+e\_{45})\wedge \Lambda^2(U/e\_1) = 0 \},
$$
so this is isomorphic to $Q^3 \subset Gr(2,4) \subset Gr(3,V)$, and
$$
f^{-1}(e\_{123})=\{ U\ |\ \dim (U \cap \langle e\_1,e\_2,e\_3 \rangle) \ge 2 \}, 
$$
which is isomorphic to a $P^3$ bundle over $P^2$ with a section contracted to a point.


