  In this paper, we address the partial regularity of suitable weak solutions
of the incompressible Navier--Stokes equations. We prove an interior regularity
criterion involving only one component of the velocity. Namely, if $(u,p)$ is a
suitable weak solution and a certain scale-invariant quantity involving only
$u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at
$(x_0,t_0)$.
