4 - Discriminants and modality of F-manifolds  pp. 40-60

By Claus Hertling

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Discriminants play a central role in singularity theory. Usually they have a rich geometry and say a lot about the mappings or other objects from which they are derived. The discriminant D of a massive F-manifold M with a generating function (cf. Definition 3. 18) is an excellent model case of such discriminants, having many typical properties.

Together with the unit field it determines the whole F-manifold in a nice geometric way. This is discussed in section 4.1 (cf. Corollary 4.6). In section 4.3 results from singularity theory are adapted to show that the discriminant and also the bifurcation diagram are free divisors under certain hypotheses.

The classification of germs of 2-dimensional massive F-manifolds is nice and is carried out in section 4.2. Already for 3-dimensional germs it is vast (cf. section 5.5). In section 4.4 the Lyashko–Looijenga map is used to prove that the automorphism group of a germ of a massive F-manifold is finite. There also the notions modality and μ-constant stratum from singularity theory are adapted to F-manifolds. In section 4.5 the relation between analytic spectrum and multiplication is generalized. This allows F-manifolds to be found in natural geometric situations (e.g. hypersurface and boundary singularities) and to be written down in an economic way (e.g. in 5.22, 5.27, 5.30, 5.32).

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