  We are concerned with the problem,originated from Seregin [18,19,20], what
are minimal sufficiently conditions for the regularity of suitable weak
solutions to the 3d Naiver-Stokes equations. We prove some interior regularity
criteria, in terms of either one component of the velocity with sufficiently
small local scaled norm and the rest part with bounded local scaled norm, or
horizontal part of the vorticity with sufficiently small local scaled norm and
the vertical part with bounded local scaled norm. It is also shown that only
the smallness on the local scaled $L^{2}$ norm of horizontal gradient without
any other condition on the vertical gradient can still ensure the regularity of
suitable weak solutions. All these conclusions improve pervious results on the
local scaled norm type regularity conditions.
