Some properties of a Brauer class

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 ${\mathrm{Pic}}_{X/k}^0$ has natural involutions. We show the class $\alpha$ splits at some height one points in ${\mathrm{Pic}}_{X/k}$.

10 pages, comments welcome