The space of morphisms $\mathbb P^1\to \mathbb P^2$ of degree $3$ is a Zariski open subset $U$ of $$ \mathbb P(H^0(\mathbb P^1, \mathcal O(3))^{\oplus 3}) \cong \mathbb P^{11} $$ the generic rational map is a nodal cubic, so the subset $V\subset U$ of cuspidal curves is of codimension at least $1$.
How can one find equations for $\overline{V}\subset \mathbb P^{11}$? I expect it to be a hypersurface but I cannot figure out how to write down the equation.