Affine holomorphic bundles over ${\mathbb{P}}_{\mathbb{C}}^1$ and apolar ideals

We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X={\mathbb{P}}_{\mathbb{C}}^1$. We study the moduli space of framed, non-degenerate rank 2 affine bundles over ${\mathbb{P}}_{\mathbb{C}}^1$ whose linearisation, viewed as locally free sheaf, is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^1}(n_1)\oplus {\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^1}(n_2)$ where $n_1>n_2$. We show that this moduli space can be identified with the “topological cokernel” of a morphism of linear spaces over the projective space $\mathbb{P}(\mathbb{C}[X_0,X_1{]}_l)$ of binary forms of degree $l:=-2-n_2$, in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1{]}_l)$ defined by the level sets of the fibre dimension map is determined explicitly by $d:=n_1-n_2$ and the cactus rank stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1{]}_l)$.