Towards transversality of singular varieties: splayed divisors

Details Towards transversality of singular varieties: splayed divisors Towards transversality of singular varieties: splayed divisors

2012-01-10

arxiv.org/abs/1201.2186v1

Mathematics

Algebraic Geometry

14 pages

Scientific paper

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.

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