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

TL;DR

This paper classifies affine holomorphic bundles over the complex projective line, linking their moduli space to binary forms and stratifications determined by rank and cactus rank, revealing geometric structures and fiberings.

Contribution

It provides a detailed description of the moduli space of affine bundles over , connecting it to the topological cokernel of morphisms and explicit stratifications based on binary form ranks.

Findings

Moduli space identified with topological cokernel over projective space.

Stratification of the moduli space explicitly determined by rank parameters.

Fibering of the moduli space over binary form projective space.

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 $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$.