Adjoint divisors and free divisors

David Mond and Mathias Schulze

Journal of Singularities
volume 7 (2013), 253-274

Received: 24 September 2011. Received in revised form: 9 April 2013.

DOI: 10.5427/jsing.2013.7n

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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 1 stable map-germ (C^n, 0) –> (C^{n+1}, 0), 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.

Author(s) information:

David Mond Mathias Schulze
Mathematics Institute Department of Mathematics
University of Warwick University of Kaiserslautern
Coventry CV47AL 67663 Kaiserslautern
United Kingdom Germany
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