A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderon-Moreno's theorem that free divisors have the property, a proof that divisors in dimension 3 with only isolated quasi-homogeneous singularities have the property, an example of a non-free divisor with non-isolated singularity having the property, an example of a divisor not having the property, and an algorithm to compute the V-filtration along a divisor up to a given order.