Let A_g be the moduli space over F_p of g-dimensional principally polarised abelian varieties (where p is a prime and g >= 1), and let S_g be the supersingular locus. We study geometric stratifications on A_g and S_g by invariants, to deduce information on arithmetic properties of points in S_g, with a particular emphasis on their automorphism groups. Oort conjectured that all generic points of S_g have automorphism group {±1}. We will present our results that settle Oort's conjecture for g=2,3,4, and for all higher even dimensions when p >= 5.