  We prove that the spacetime singular set of any suitable Leray-Hopf solution
of the surface quasigeostrophic equation with fractional dissipation of order
$0< \alpha < \frac{1}{2}$ has Hausdorff dimension at most
$\frac{1}{2\alpha^2}\,.$ This result improves previously known dimension
estimate established in [6] and builds on the excess decay result and the
control on the particle flow already developed there. The improvement lies in
the initial iteration of the local energy inequality in analogy with the
celebrated result of Caffarelli-Kohn-Nirenberg [2] for the Navier-Stokes
equations.
