An $\text{SL}(3,\mathbb{C})$-equivariant smooth compactification of rational quartic plane curves

Let ${\mathbf{R}}_d$ be the space of stable sheaves $F$ which satisfy the Hilbert polynomial $\chi (F(m))=dm+1$ and are supported on rational curves in the projective plane ${\mathbb{P}}^2$. Then ${\mathbf{R}}_1$ (resp. ${\mathbf{R}}_2$) is isomorphic to ${\mathbf{R}}_1\cong {\mathbb{P}}^2$ (resp. ${\mathbf{R}}_2\cong {\mathbb{P}}^5$). Also it is very well-known that ${\mathbf{R}}_3$ is isomorphic to a ${\mathbb{P}}^6$-bundle over ${\mathbb{P}}^2$. In special ${\mathbf{R}}_d$ is smooth for $d\le 3$. But for $d\ge 4$ case, one can imagine that the space ${\mathbf{R}}_d$ is no more smooth because of the complexity of boundary curves. In this paper, we obtain an $\mathrm{S}\mathrm{L}(3,\mathbb{C})$-equivariant smooth resolution of ${\mathbf{R}}_4$ for $d=4$, which is a ${\mathbb{P}}^5$-bundle over the blow-up of a Kronecker modules space.