  We prove geometrically improved version of Prodi-Serrin type blow-up
criterion. Let $v$ and $\omega$ be the velocity and the vorticity of solutions
to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb
R^3\times (0, T)$. If $\left\{\left( v \times \frac{\omega}{|\omega|}
\right)\cdot \frac{\Lambda^{\beta}v}{|\Lambda^{\beta}v|}\right\}_+ \in
L^{\gamma, \alpha}_{x,t} (Q_T)$ with $3/\gamma +2/\alpha \leq 1$ for some
$\gamma >3$ and $1 \leq \beta \leq 2$, then the local smooth solution $v$ of
the Navier-Stokes equations on $(0,T)$ can be continued to $(0, T+\delta)$ for
some $\delta >0$. We also prove localized version of a special case of this.
Let $v$ be a suitable weak solution to the Navier-tokes equations in a
space-time domain containing $z_0= (x_0, t_0)$, let $Q_{z_0, r}=B_{x_0, r}
\times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if
either $\left\{\left( v \times \frac{\omega}{|\omega|}\right) \cdot
\frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma,
\alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 1$, or
$\left\{\left(\frac{v}{|v|} \times \omega\right) \cdot \frac{\nabla \times
\omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma,
\alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 2$,
($\gamma \geq 2$, $\alpha \geq 2$), then $z_0$ is a regular point for $v$. This
improves previous local regularity criteria for the suitable weak solutions.
