# Some properties of a Brauer class

## TL;DR

This paper investigates the properties of a Brauer class associated with a smooth proper curve over a field, revealing that the related division algebra has natural involutions and that the class splits at certain points.

## Contribution

It demonstrates the existence of natural involutions on the division algebra linked to the Brauer class and shows the class splits at specific height one points in the Picard scheme.

## Key findings

- Division algebra on Pic^0_{X/k} has natural involutions
- Brauer class splits at some height one points
- Obstruction to Picard functor representability

## Abstract

Let $X$ be a smooth proper curve defined over a field $k$. The representability of the relative Picard functor is obstructed by a class $\alpha\in\mathrm{Br}(\mathrm{Pic}_{X/k})$. We show the associated division algebra on $\mathrm{Pic}^0_{X/k}$ has natural involutions. We show the class $\alpha$ splits at some height one points in $\mathrm{Pic}_{X/k}$.