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

Abstract

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.