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 \(d\leq 3\). But for \(d\geq4\) 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 \(\mathrm{SL}(3,\mathbb{C})\)-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.