  We show that the norm of the Riemann curvature tensor of any smooth solution
to the Ricci flow can be explicitly estimated in terms of its initial values on
a given ball, a local uniform bound on the Ricci tensor, and the elapsed time.
This provides a new, direct proof of a result of Sesum, which asserts that the
curvature of a solution on a compact manifold cannot blow up while the Ricci
curvature remains bounded, and extends its conclusions to the noncompact
setting. We also prove that the Ricci curvature must blow up at least linearly
along a subsequence at a finite time singularity.
