Towards transversality of singular varieties: splayed divisors

TL;DR

This paper introduces the concept of splayed divisors, a generalization of transversally intersecting smooth hypersurfaces, and provides characterizations, tests, and properties related to their geometry and algebraic structure.

Contribution

It defines splayed divisors, characterizes them via Jacobian ideals and logarithmic derivations, and explores their geometric and algebraic properties, including normal crossings and Hilbert-Samuel polynomial additivity.

Findings

A Jacobian ideal property characterizes splayed divisors.

Union of smooth hypersurfaces with normal crossings is a free divisor with radical Jacobian ideal.

Hilbert-Samuel polynomials of splayed divisors exhibit additivity.

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.