  S. Kond\=o used periods of $K3$ surfaces to prove that the moduli space of
genus three curves is birational to an arithmetic quotient of a complex 6-ball.
In this paper we study Heegner divisors in the ball quotient, given by
arithmetically defined hyperplane sections of the ball. We show that the
corresponding loci of genus three curves are given by: hyperelliptic curves,
singular plane quartics and plane quartics admitting certain rational
"splitting curves".
