Mass formula and Oort's conjecture for supersingular abelian threefolds

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$.