A complete description of the cohomological invariants of even genus hyperelliptic curves

TL;DR

This paper fully characterizes the mod 2 cohomological invariants of even genus hyperelliptic curves over various fields, providing explicit descriptions and analyzing their algebraic structure, with implications for compactifications and Picard groups.

Contribution

It extends the computation of cohomological invariants to non algebraically closed fields and describes their multiplicative structure for even genus hyperelliptic curves.

Findings

Cohomological invariants of $ar{ ext{H}}_g$ are trivial.

Provides explicit functorial description of invariants.

Extends results to positive characteristic.

Abstract

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic