Adjoint divisors and free divisors

TL;DR

This paper explores conditions under which adding the adjoint divisor to certain divisors results in free divisors, revealing new examples and suggesting a broader underlying phenomenon in divisor theory.

Contribution

It identifies specific scenarios where the adjoint divisor makes a divisor free, including new examples and a general phenomenon in divisor stability and versality.

Findings

Adding the adjoint divisor can produce free divisors in certain cases.

New examples of free divisors are constructed from stable germs and versal deformations.

A foundational result is provided that generates numerous new free divisor examples.

Abstract

We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both also involve stability or versality. In the first, D is the image of a corank one stable germ of a map from complex n-space to complex (n+1)-space, and is not free. In the second, D is the discriminant of a versal deformation of a weighted homogeneous function with isolated critical point (subject to certain numerical conditions on the weights). Here D itself is already free. We also prove an elementary result, inspired by these first two, from which we obtain a plethora of new examples of free divisors. The presented results seem to scratch the surface of a more general phenomenon that is still to be revealed.