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We prove geometrically improved version of Prodi–Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier–Stokes equations and denote { f } + = max { f , 0 } , Q T = R 3 × ( 0 , T ) . If { ( v × ω ∣ ω ∣ ) ⋅ Λ β v ∣ Λ β v ∣ } + ∈ L x , t γ , α ( Q T ) with 3 / γ + 2 / α ≤ 1 for some γ > 3 and 1 ≤ β ≤ 2 , then the local smooth solution v of the Navier–Stokes equations on ( 0 , T ) can be continued to ( 0 , T + δ ) for some δ > 0 . We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier–Stokes equations in a space–time domain containing z 0 = ( x 0 , t 0 ) , let Q z 0 , r = B x 0 , r × ( t 0 − r 2 , t 0 ) be a parabolic cylinder in the domain. We show that if either { ( v × ω ∣ ω ∣ ) ⋅ ∇ × ω ∣ ∇ × ω ∣ } + ∈ L x , t γ , α ( Q z 0 , r ) with 3 γ + 2 α ≤ 1 , or { ( v ∣ v ∣ × ω ) ⋅ ∇ × ω ∣ ∇ × ω ∣ } + ∈ L x , t γ , α ( Q z 0 , r ) with 3 γ + 2 α ≤ 2 , ( γ ≥ 2 , α ≥ 2 ), then z 0 is a regular point for v . This improves previous local regularity criteria for the suitable weak solutions. [ABSTRACT FROM AUTHOR]