  Let $p$ be a prime number and let $k$ be an algebraically closed field of
characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative
group scheme which arises as the kernel of $p$ on a $p$-divisible
(Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over
$k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian
of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with
polarizations. Our main tools are the Kraft classification of $BT_1$ group
schemes, a theorem of Oda, and a combinatorial description of the de Rham
cohomology of Fermat curves.
