Stable Cohomology of Discriminant Complements for an algebraic curve

TL;DR

This paper proves that, within a stable range, the cohomology of the space of algebraic sections of a line bundle on a curve matches that of smooth sections, revealing a deep topological equivalence.

Contribution

It establishes an isomorphism between algebraic and smooth section cohomologies for line bundles on curves in a stable range, bridging algebraic and differential topology.

Findings

Cohomology of algebraic sections matches smooth sections in a stable range.

Provides a topological equivalence between algebraic and smooth section spaces.

Advances understanding of the topology of discriminant complements for algebraic curves.

Abstract

In this paper we show that in a stable range the cohomology of the space of regular algebraic sections of a line bundle $\mathscr{L}$on a curve $X$ is isomorphic to the cohomology of the space of regular $C^{\infty}$sections of the same line bundle.