Holomorphic pre-symplectic form on the nested Hilbert scheme $\mbox{Hilb}^{3,4}(\mathbb{C}^2)$

We regard the pre-hyperkähler structure on the moduli space of framed flags of sheaves $\mathcal{F}(1,3,1)$ on the projective plane $\mathbb{P}^2$ via an adaptation of the ADHM construction of framed sheaves. Then, we study and categorize the degenerate points of the holomorphic pre-symplectic form presented in the moduli space.