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

Abstract

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 $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2 affine bundles over $\mathbb{P}^1_\mathbb{C}$ whose linearisation, viewed as locally free sheaf, is isomorphic to $ {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_1)\oplus {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(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 $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$.