Vector fields and a family of linear type modules related to free divisors

This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered.