Towards Transversality of Singular Varieties: Splayed Divisors

Abstract

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called splayed divisors. A splayed divisor is characterized by a property of its Jacobian ideal. This yields an effective test for splayedness. Two further characterizations of a splayed divisor are shown, one reflecting the geometry of the intersection of its components, the other one using K. Saito's logarithmic derivations. As an application we prove that a union of smooth hypersurfaces has normal crossings if and only if it is a free divisor and has a radical Jacobian ideal Further it is shown that the Hilbert–Samuel polynomials of splayed divisors have the natural additivity property.