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 satisfy a certain additivity property.

15 pages, 1 figure; v2: minor revision: inaccuracies corrected and references updated; v3: final version, to appear in Publ. RIMS

References in corpus (2)

• Splayed divisors and their Chern classes
• Characterizing normal crossing hypersurfaces

Cited by in corpus (6)

• Splayed divisors and their Chern classes
• Characterizing normal crossing hypersurfaces
• Chern classes of splayed intersections
• Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors
• On Saito's normal crossing condition
• Crepant resolutions of double covers: On the Cynk-Hulek criterion for crepant resolutions of double cover