Every $BT_1$ group scheme appears in a Jacobian

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